Math AA SL

274 Core SL content: Review Exercise

9 ABCDV is a solid glass pyramid. The base of the pyramid is a square of side 3.2 cm. The vertical
height is 2.8 cm. The vertex V is directly above the centre O of the base.
V

C
D

O

AB

a Calculate the volume of the pyramid.
b The glass weighs 9.3 grams per cm3 . Calculate the weight of the pyramid.
c Show that the length of the sloping edge VC of the pyramid is 3.6 cm.
d Calculate the angle at the vertex ˆ BVC.
e Calculate the total surface area of the pyramid.

Mathematical Studies SL November 2007 Paper 2 Q2

10 a Sketch the graph of the function f(x) = 2x3+ , for 1−0 x 10, indicating clearly the axis
intercepts and any asymptotes. x+4

b Write down the equation of the vertical asymptote.

c On the same diagram sketch the graph of g(x) = x + 0.5.
d Using your graphical display calculator write down the coordinates of one of the points of

intersection on the graphs of f and g, giving your answer correct to five decimal places.

e Write down the gradient of the line g(x) = x + 0.5.

f The line L passes through the point with coordinates (-2, -3) and is perpendicular to the line
g(x). Find the equation of L.

Mathematical Studies May 2008 Paper 2 TZ1 Q1

11 Find the equation of the normal to the curve y x= x+at 3th2e point where x = -2. Give your
answer in the form ax b+y+c = 0 .

12 a Find the value of x given that log 10 x = 3.
b Find log 10 0.01.

13 a Expand and simplify (2 )x(x3 − 3−x) .5
b Differentiate y = (2 x) (x3 − 3−x) .5

Core SL content: Review Exercise 275

14 A geometric series has first term 18 and common ratio r.

a Find an expression for the 4th term of the series.

b Write down the formula for the sum of the first 15 terms of this series.

c Given that the sum of the first 15 terms of the series is 26.28 find the value of r.

15 Theo is looking to invest £200 for 18 months. He needs his investment to be worth £211 by the

end of the 18 months. What interest rate does he need? Give your answer as a percentage correct

to one decimal place. k for X = 1, 2, 3, 4.
16 The distribution of a discrete random variable X is given by ( P X x= =) x2

Find E(X).

17 A box holds 240 eggs. The probability that an egg is brown is 0.05.
a Find the expected number of brown eggs in the box.
b Find the probability that there are 15 brown eggs in the box.
c Find the probability that there are at least 10 brown eggs in the box.

Mathematics SL May 2011 Paper 2 TZ1 Q5

18 The vertices of quadrilateral ABCD as shown in the diagram are A(3, 1), B(0, 2), C(-2, 1) and

D(-1, -1).
y

4

2B A
C

−4 −2 24 x

D− 2

−4

a Calculate the gradient of line CD.
b Show that line AD is perpendicular to line CD.
c Find the equation of line CD. Give your answer in the form ax + by = c where a, b, c ∈ .
Lines AB and CD intersect at point E. The equation of line AB is x + 3y = 6.
d Find the coordinates of E.
e Find the distance between A and D.
The distance between D and E is 20.
f Find the area of triangle ADE.

Mathematical Studies May 2009 Paper 2 TZ1 Q3

276 Core SL content: Review Exercise

19 a Jenny has a circular cylinder with a lid. The cylinder has height 39 cm and diameter 65 mm.
i Calculate the volume of the cylinder in cm3 . Give your answer correct to two decimal
places.
The cylinder is used for storing tennis balls.
Each ball has a radius of 3.25 cm.
ii Calculate how many balls Jenny can fit in the cylinder if it is filled to the top.
iii I Jenny fills the cylinder with the number of balls found in part ii and puts the lid on.
Calculate the volume of air inside the cylinder in the spaces between the tennis balls.
II Convert your answer to iii I into cubic metres.

b An old tower (BT) leans at 10° away from the vertical (represented by line TG).
The base of the tower is at B so that MBˆ T = 100°.
Leonardo stands at L on flat ground 120 m away from B in the direction of the lean.
He measures the angle between the ground and the top of the tower T to be ˆ BLT = 26.5°.
T

100° 26.5°

M MB = 200 BG BL = 120 L

i I Find the value of angle ˆBTL.

II Use triangle BTL to calculate the sloping distance BT from the base B to the top, T of
the tower.

ii Calculate the vertical height TG of the top of the tower.
iii Leonardo now walks to point M, a distance 200 m from B on the opposite side of the

tower. Calculate the distance from M to the top of the tower at T.

Mathematical Studies SL May 2007 Paper 2 Q2

Core SL content: Review Exercise 277

20 The following table shows the number of bicycles, x, produced daily by a factory and their total
production cost, y, in US dollars (USD). The table shows data recorded over seven days.

Number of bicycles, x Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 21
Production cost, y 12 15 14 17 20 18

3900 4600 4100 5300 6000 5400 6000

a i Write down the Pearson’s product–moment correlation coefficient, r, for these data.
ii Hence comment on the result.

b Write down the equation of the regression line y on x for these data, in the form y = ax + b.
c Estimate the total cost, to the nearest USD, of producing 13 bicycles on a particular day.

All the bicycles that are produced are sold. The bicycles are sold for 304 USD each.
d Explain why the factory does not make a profit when producing 13 bicycles on a particular day.
e i Write down an expression for the total selling price of x bicycles.

ii Write down an expression for the profit the factory makes when producing x bicycles on a
particular day.

iii Find the least number of bicycles that the factory should produce, on a particular day, in
order to make a profit.

Mathematical Studies May 2015 Paper 2 TZ2 Q6

21 It is known that, among all college students, the time taken to complete a test paper is normally
distributed with mean 52 minutes and standard deviation 7 minutes.
a Find the probability that a randomly chosen student completes the test in less than 45 minutes.
b In a group of 20 randomly chosen college students, find the probability that:
i exactly one completes the test in less than 45 minutes
ii more than three complete the test in less than 45 minutes.

22 In a class of 26 students, 15 study French, 14 study biology and 8 study history.

Of those students, 7 study both French and biology, 4 study French and history, and 3 study
biology and history.
a Using a Venn diagram, or otherwise, find how many students study all three subjects.
b Find the probability that a randomly selected student studies French only.
c Given that a student studies French, what is the probability that they do not study biology?
d Two students are selected at random. What is the probability that at least one of them studies

history?

11 Analysis and approaches: Proof

ESSENTIAL UNDERSTANDINGS

n Number and algebra allow us to represent patterns, show equivalences and make generalizations.
n Algebra is an abstraction of numerical concepts and employs variables to solve mathematical

problems.

In this chapter you will learn...

n about equations and identities
n how to construct simple deductive proofs.

CONCEPTS

The following key concepts will be addressed in this chapter:
n Numbers and formulae can appear in different but equivalent forms, or

representations, which can help us to establish identities.
n Formulae are a generalization made on the basis of specific examples, which can

then be extended to new examples.

PRIOR KNOWLEDGE

Before starting this chapter, you should already be able to complete the following:

1 Expand and simplify (2 x1)−( 4).x +

2 Evaluate 3 2 .
+4 5

n Figure 11.1 Does the word 'proof' mean different things in different contexts?

Analysis and approaches: Proof 279

Proof is one of the most important elements of mathematics as it allows us to establish
whether certain results that we might suspect are true, really are true.

In science, reasoning is often inductive; scientists base their theories on observation but
have to be open to revising their ideas if new evidence comes to light.

In mathematics, deductive proof starts with a statement you know to be true and
proceeds with a sequence of valid steps to arrive at a conclusion. As long as the starting
point is true and you do not make any subsequent mistakes in working from one step to
the next, you know for certain that the conclusion is true.

TOK Links

Look at the pictures in Figure 11.1. Does ‘proof’ mean the same in different disciplines? What
is the difference between proof in maths, science and law? To what extent can we talk about
proof in relation to philosophy, or social sciences?

Starter Activity

Necklaces are made by threading n gold or silver beads onto a closed loop. For example, if
n 2=, there are three possible necklaces that could be made: SS, GG, GS.
a Find how many different necklaces can be made for

i n 0=
ii n 1=
iii n 3=
b On the basis of these results (including = n 2), suggest a formula for the number of
necklaces that can be made with n beads.

c Check your formula with the result for = n 4.

LEARNER PROFILE – Communicators

Is a proof the same as an explanation?

280 11 Analysis and approaches: Proof

11A The structure of mathematical proof

n The symbols and notation for equality and identity

Equations are one of the most frequently used mathematical structures. An equation
is only true for some particular value(s) of a variable. For example, x2 = 4 is true for
x = ±2 but not for any other values of x.

An identity is true for all values of a variable. To emphasize that we are dealing with
an identity and not just an equation, the symbol ≡ is used instead of the = symbol.
For example, x2 − ≡4 (− x 2)(x2)+.

Tip WORKED EXAMPLE 11.1

The identity symbol, ≡, x2 + 6+x2≡(+) + x a b 2
Find the values of the constants a and b.
is often only used for
emphasis. You will find The expression on the left-hand side x2x+ 6+2≡(+) + x a b 2 2
lots of identities are (LHS) must be identical to that on the ≡ +x2a+x+a2 b
written with an equals
sign instead. right-hand side (RHS) for all values
of x. Multiply out the brackets so
you can compare the two sides Coefficient of x: =6 a2 , so = a 3
Constant term: 2 = +a 2b
The coefficient of x (the number in front of x) So, 2 3=2+ b
must be the same on both sides b = −7

The constant term must be
the same on both sides

You already know a 3=

CONCEPTS – REPRESENTATIONS

Are the expressions x2 + +6x2and x ( 3)+7− 2identical? Why might you use one rather
than the other? You will find out more about this in Chapter 16.

In Worked Example 11.1 we compared coefficients, but it is important to note that

just because two expressions represent the same value, they do not have to have

comparable parts. For a = 1 it does not mean that a must equal 1 and b
must equal 2. example, ibf 2

11A The structure of mathematical proof 281

n Simple deductive proof

To prove that a statement is true, you need to start with the given information and
proceed by a sequence of steps to the conclusion.

Often proofs involve algebra as this is a tool that enables us to express ideas in general
terms.

WORKED EXAMPLE 11.2

Show that 1 = 1 1 .
−3 4 12

We can start from one side and show it is 1 − =14 × 41 − 3×3 1
equivalent to the other side. Normally starting 3 43 4
from the more complicated side – in this case
43
the LHS – and simplifying is easiest. Create ≡ 1−2 12
a common denominator of 12 by multiplying
43
the first fraction by 4 and the second by 3 ≡ −12

Combine the fractions and simplify ≡ 112

TOK Links

Would you use this proof to explain this fact to someone just meeting fractions? Would you
need to include every step if you were proving it to a university professor? Whenever you are
communicating knowledge, you must consider the audience. However, sometimes in maths,

‘obvious’ facts turn out to be very difficult to prove. For example, the ‘fact’ that parallel lines
never meet has been challenged by some geometers. The philosopher Bertrand Russell spent
many pages proving tha1t 1+2=. Whenever we are producing a proof we need to know
what the accepted facts are in our mathematical community.

282 11 Analysis and approaches: Proof

Tip WORKED EXAMPLE 11.3

In questions, the Prove that 6 a5 + − +4≡a3 1 .
command terms 3 2 6
‘prove’ and ‘show that’
both just mean you In this case the LHS is more 6a5+ − 4a3+ ≡ 2(6 a5)+ − 3(4a3)+
should set out your complicated so we start there and 3 2 23 32
argument in a clear and try to show it is equivalent to the
logical way. RHS. Create a common denominator

of 6 by multiplying the first 12 a10+ 12a9+
fraction by 2 and the second by 3 6 6

≡ −

Combine the fractions and ≡ 12a10+1−2 −9 a
simplify the algebra 6

1
≡6

WORKED EXAMPLE 11.4

Prove that the sum of an even number and an odd number is always odd.

Define a general even number... Let the even number be 2n, for some n ∈ .
Let the odd number be 2m + 1,
...and a general odd number for some m ∈ .
(you could also use 2m - 1)
Don’t use 2n + 1 (or 2n - 1) as this 2n(2m+1+) 2=2+1+ n m
would be the next integer up (or down) = +2(+n) m1
from 2n, which would be too specific = 2k + 1, for some k ∈

Make it clear that the result is So, the sum is odd.
an odd number by writing the

result in the form 2k + 1

Make a conclusion

11A The structure of mathematical proof 283

CONCEPTS – GENERALIZATION

When solving a problem, lots of people start by trying some cases in order to convince
themselves about what is happening. While this is a good initial tactic, no matter
how many cases you try you will never create a mathematical proof (unless you can
also argue that you have covered all possible cases). For example, there is a famous
conjecture, due to Goldbach, that states:

‘Every odd number bigger than 1 is either prime or can be written as a prime plus twice
a square number.’

This does seem to work for a lot of the odd numbers – mathematicians have used
computers to search for counterexamples up to 1013 and only found two: 5777 and
5993.

Another famous conjecture by Goldbach states that every even integer greater than
2 can be written as the sum of two primes. This has been tested up to 1018 and no
counterexamples have been found, but this does not mean mathematicians believe it
has been proven.

The big advantage of algebra is that it is not considering specific examples, but is used
to represent all possible examples. This is what makes it so useful in proof.

Be the Examiner 11.1

Prove that the product of an even and an odd number is always even.
Which is the correct solution? Identify the errors made in the incorrect solutions.

Solution 1 Solution 2 Solution 3

2n(2n1)+2(=2 n2n+ ) 2m(2n1) +2(2= mn +m ) If n is even, then +n 1 is
odd or vice versa.
= 2k = 2k So, n(n1)+2= k
So, the product is even. So, the product is even. So, the product is even.

284 11 Analysis and approaches: Proof

Exercise 11A

1 Given that + x+2 ≡ +8x23 ( x a )2 + b, find the values of a and b.
2 Given that −x2 12x1−( ≡ − x a )2 − b, find the values of a and b.
3 Find the values of p and q such that (2 x5)−(2 5) x + ≡ px2q2− .

4 Prove that (n1)− 2 + n+2 + (n1) 2 ≡ 3n2 + 2.

5 Prove that x 3 2− − − 2≡x3 5 .
3 2 6

6 Prove that x 7 6+ − +3≡x5 17 x .
12 10 60

7a Show that 1 2 + 14= 3.
4
b Show that the algebraic generalization of this is n n1 n1+ 2≡ 3
2 .

8 Which of the following equations are, in fact, identities?
a (x − 5)(x + 5=) x2 − 25

b ( x5)+ 2 = x+2 25

c (x − 5)2 = x2 − 10x + 25 x3 − 5x2 + 3+x9≡ + (x)a( x)b − 2.
9 Find the values of a and b so that

10 Find the values of p and q such that x3 − 2x2 + 5−x1≡0 + (x2p x q)( )− .
11 5
11 Prove that −x 2 − +x 3 ≡ x2 + −x 6.

11 2 .
12 a Show that − 3= 5 15
b Show that the algebraic generalization of this is − n 1 1 − n+1 1 ≡ −n22 1.

13 Dina says that x y3 − =3− (x)y x ( 2 + y2) is an identity.

a Find one example to show that Dina is wrong.

b Find the value of a such that x y3 − ≡3− (x)y x a(xy2 +y + 2).

14 Prove that n(2 1)+(2 21)− n + is an even number for all positive integer values of n.

15 Prove that n(2 1)+(221)− n − is2a multiple of 8 for all positive integer values of n.

16 Prove algebraically that the difference between the squares of any two consecutive integers is always an odd
number.

17 Prove that the sum of the squares of two consecutive odd numbers is never a multiple of 4.

Checklist

n You should be able to work with and understand the notation for identities.
An identity occurs when two expressions always take the same value for all allowed values of the variable.
When this is emphasized the symbol ≡ is used in place of the equals sign.
If two polynomials are identical you can compare coefficients.

n You should be able to construct simple deductive proofs.
The usual method is to start from one side and show through a series of identities that it is equivalent to the
other side.

Mixed Practice 285

n Mixed Practice

1 Which of the following are identities?
a x ( 2+) 3 = x+3 8
b x 3 + 6=4+( 4)x( x2 − 4+x16)
c x 4 − 4= ( x2 − 2)(x2 + 2)

2 a Use an example to show that x 2 + 5=(+1)x( 5) x + is not an identity.
b Find the value of k so that x kx2 + + ≡ 5+ ( 1)x( 5). x +

3 a Use an example to show that x 2 + 4+x9=( +3) x 2 is not an identity.
b Given that x 2 + 4+x9≡(+ x p )2 + q, find the values of p and q.

4 Find the values of a and b such that x 2 − 8≡x−x a( )2 − b.

5 Prove that

x − 2− − x≡ 3 x .
23 6

6 Find the values of A and B such that

A(x2+) + B(x2 1)−5≡. x

7 Prove that n(2 3)+(2 23−) n − 2 is a multiple of 12 for all positive integer values of n.

8 Prove algebraically that the difference between any two different odd numbers is an even number.

9 The product of two consecutive positive integers is added to the larger of the two integers. Prove that
the result is always a square number.

10 Prove that( x x − x− 1 2 ≡ −( x 2 .
− 2)2 2)2

11 Prove that the sum of the squares of any three consecutive odd numbers is always 11 more than a
multiple of 12.

12 a Find the values of a and b such that − x=3 − 8 ( 2)x( x2a+x b+ . )
b n is a positive integer greater than 3. Prove that − 8n3 is never a prime number.

Analysis and approaches:

12 Exponents and logarithms

ESSENTIAL UNDERSTANDINGS

n Number and algebra allow us to represent patterns, show equivalences and make generalisations.
n Algebra is an abstraction of numerical concepts and employs variables to solve mathematical problems.

In this chapter you will learn...

n how to extend the laws of exponents to general rational exponents
n how to work with logarithms to any (positive) base
n how to use the laws of logarithms
n how to change the base of a logarithm
n how to solve exponential equations.

CONCEPTS

The following key concepts will be addressed in this chapter:
n Numbers and formulae can appear in different but equivalent forms, or

representations, which can help us to establish identities.
n Patterns in numbers inform the development of algebraic tools that can be applied

to find unknowns.

PRIOR KNOWLEDGE

Before starting this chapter, you should already be able to complete the following:

1 Simplify the following: b 8c2d3 c (3a)4b2−2
a 3x2 y4 × 5x7 y 2c3d

Find the value of y for which log ( 9y) +2.

1 x 10
9
)2 =

.

43 SSoollvvee tthhee eeqquuaattiioonn e32xx-5-1==( 11.

n Figure 12.1 Why are logarithms used to describe the severity of an earthquake, the volume of a sound, the growth of
bacteria and the acidity of a solution?

Analysis and approaches: Exponents and logarithms 287

You have already seen how the laws of exponents allow you to manipulate exponential
expressions and how this can be useful for solving some types of exponential equation.

In the same way, it is useful to have some laws of logarithms that will enable you to
solve some more complicated exponential equations and equations involving different
logarithm terms. It should be no surprise, given the relationship between exponents and
logarithms, that these laws of logarithms follow from the laws of indices.

Starter Activity

Look at the pictures in Figure 12.1. Investigate how logarithmic functions are used to
measure or model these different phenomena.

Now try this problem:

By trying different positive values of x and y, suggest expressions for the following in
terms of ln x and ln y.
a ln(xy)

( )blnx
y

c ln(xy )

Do your suggested relationships work for log 10 as well?

LEARNER PROFILE – Reflective

Is being really good at arithmetic the same as being really good at mathematics?

288 12 Analysis and approaches: Exponents and logarithms

For a 12A Laws of exponents with rational
reminder exponents
of the laws
of exponents for In order to extend the laws you know for integer exponents to non-integer rational
integer exponents, exponents, you need a new law.
see Section 1A.
KEY POINT 12.1
Tip
a 1n = n a
If n is even then an is
defined to be positive, Proof 12.1 n a is the number which equals a when
and it is only real if
a is also positive. If n 1
is odd, then a can be
negative, and an has Explain why a n = n a .
the same sign as a.
We need a defining feature
of na raised to the exponent n.

Use the fact that x( )a b ab≡ x ( )We know that a⎛ 1n⎞ n1 n ≡ a1

⎝⎜ ⎠⎟ ≡ an

Therefore a 1n has the defining property of a n

CONCEPTS – REPRESENTATION

You might ask why writing the same thing in a different notation has any benefit. Using
an exponent representation of na has distinct advantage as it allows us to use the
laws of exponents on these expressions.

WORKED EXAMPLE 12.1

1 1

Evaluate 16 4 . 16 416= 4
=2
Use a 1n = n a

Combining the law in Key Point 12.1 with the law that states that( )am≡an mn allows us
to cope with any rational exponent.

12A Laws of exponents with rational exponents 289

WORKED EXAMPLE 12.2 ( )27−2= 1 −2
3
Evaluate 27 −23. 27 3
Use (a )m n ≡ a mn to split
the exponent

Remember that a negative 1

exponent turns into 1 divided = ( 3)27 2

by the same expression with a
1
positive exponent. = 3 27
27 3

You should know small perfect = =312 1
squares and cubes. You can 9
recognise that 237 3 =

You will also need to be able use this new law in an algebraic context.

WORKED EXAMPLE 12.3 4x4 x
3x = 1
Write x4 in the form kx n.
3x x3
Use a1n = n a on the
denominator = 4x1−13

Then use aamn = −aman 2

= 4x3

Exercise 12A

For questions 1 to 3, use the method demonstrated in Worked Example 12.1 to evaluate the expression

without using a calculator. 1 1
1
1 a 49 2 2a8 3 3 a 256 4

1 1 1

b 25 2 b 27 3 b 625 4

For questions 4 to 8, use the method demonstrated in Worked Example 12.2 to evaluate the expression

without using a calculator. 3 6 a 100 − 1
2
2 5 a 625 4

4a8 3

3 2 b 1000 − 1
3
b 16 4 b 125 3

7a8 − 2 8 a 32 − 2
3 5

b 27 − 2 b 100000 − 3
3 5

For questions 9 to 13, use the method demonstrated in Worked Example 12.3 to simplify each expression,

giving your answers in the form ax p. x2 4x
3x x2
9 a x2x 10 a 11 a

b x3x x2 b 5x
bx x3

290 12 Analysis and approaches: Exponents and logarithms

12 a x2 13 a 3 x x
b 5x b x 3x2

x

3 3x

14 Find the exact value of 8 −34. 15 Find the exact value of 1 ⎛ ⎞−21.
⎝4⎠

3

16 Find the exact value of 4 ⎛⎞ 2. 17 Write x3 2 4 x in the form xp .
⎝9⎠

18 Write x33 2+ x in the form axp + bxq . 19 Write 1
3 3 xin the form axp .

20 Solve the equation x 31 21 Solve the equation x −1 2.
2 = .8 5
2=

22 Write in the form xp : 23 Write in the form xp :
xx x
3x
x2x
( )24 Write x 3 x 2 in the form xk .
25 Write ⎛ 1 3 in the form axp .
⎝2
⎞
x⎠

26 Write in the form xa + xb : 27 Write in the form xa - xb :
x2 + x
xx (x +x x x)( − )

28 Write in the form axk : x
1
29 Write in the form axp + bxq :
3x x x2 + 3 x
30 Given that y = 2 32 x , write y4 in the form axk . 2x

32 Given that x = 3 y, write y in the form xk . 31 Given that y = 27 x, write y3 in the form axk .

33 Given that y = 3 2 , write y3 in the form axk .
x

34 Given that y x =x , express x in terms of y. 2
36 Solve the equation x = 2 3 x.
35 Solve the equation x 3 = .9

12B Logarithms

In Section 1C, you worked with logarithms to base 10 and logarithms to base e. Now
you need to be able to work with any (positive) base.

The result WORKED EXAMPLE 12.4

from Key Calculate the value of log 1 .

Point 1.4 3 9

holds for any

(positive) base: Let the value of log 319 be x x l=og 1 3 9

a b=xis equivalent
log a x
to = .
b
1
Use a b= x is equivalent Therefore, 3x = 9
to log b=a x
log 1 = −2
You can see (or experiment 39
to find) that x = −2

12B Logarithms 291

WORKED EXAMPLE 12.5 log53(y1)− = −

Find the exact value of y if log5( 3y)−1= − .
Use a b=xis equivalent
to log ba=x

y −3 5= = −1 1
5

y = 156

n Laws of logarithms

The laws of exponents lead to a set of laws of logarithms.

KEY POINT 12.2

l logb xy = logb x + logb y
x
l logb y = logb x − logb y

l logb xm = m logb x
where b ,x, y > 0

Tip CONCEPTS – PATTERNS

Be careful not to apply Many rules like the ones in Key Point 12.2 are explored and discovered by systematically
invented, similar-looking looking at patterns in numbers.
rules of logarithms.
For example, many For example: log1200 1.3=0103 log12000 2.=30103 log120000 3.=30103
students claim that log120 0.=30103
log( x +y =) log x + log y
or that The first law of logarithms is proved below in Proof 12.2. The others can be proved
similarly from the corresponding law of exponents.
log( x +y =) log x log .y
Neither of these are true, Proof 12.2
in general.

Prove that logb xy = logb x + logb y. blogb x blogb y = blogbloxg+ b y
xy =b b xlobg ylog+
Start by using the related law
of exponents b mbn× b=m n+
with m = log bx and n = log by

Use blogb=x onxboth terms
of the product on the LHS

Take log b of both sides. logb xy = logb blogbloxg+ b y
Use log bbxx= on the RHS logb xy = logbloxg+ b y

292 12 Analysis and approaches: Exponents and logarithms

CONCEPTS – EQUIVALENCE

The ability to go easily between representing equations using logs and using exponents
allows us to turn our old rules into equivalent new rules. This is a very common and
powerful technique in many areas of mathematics, for example, looking at how rules in
differentiation apply to integration.

WORKED EXAMPLE 12.6

If =p log a and =qlog b, express log ⎛ a3⎞ in terms of p and q.
⎝⎜b ⎠⎟

Use logb x = logb x − logb y log ⎛ a3⎞ log alo3g− b
y ⎝⎜b ⎠⎟ =

Then use loognb xm =firmstltoegrbmx = 3log alobg−
the

Now replace log a with = −3p q
p and log a with q

One common application of the laws of logarithms is in solving log equations. The
usual method is to combine all log terms into one.

WORKED EXAMPLE 12.7

Solve log (3 x − 2−) log ( 2 x − =4) 3 .
2

Combine the log terms using log2(3 2x)−lo−g ( 4) 32 x − =
x
logb x − logb y = logb y log23⎝⎛2xx−−4 ⎞ 3
⎠
=

Remove the log 3x2− = 23
using log b a = x is x−4
equivalent to a = bx
3x2−8(=4−) x
Solve for x 3x2−8 =32− x

5x30=
x=6

Be the Examiner 12.1

Solve log ( x + 10) l+og 2 2 =.
10
10

Which is the correct solution? Identify the errors made in the incorrect solutions.

Solution 1 Solution 2 Solution 3

log1(0 1x2)+2 = log1(02 1x0)+2 = log1(02 2x0)+2=
x + 1=2 100 2x10+ 1=00 2x20+ 1=00
x = 88 x = 45 x = 40

12B Logarithms 293

n Change of base of a logarithm

The laws of logarithms only apply when the bases of the logarithms involved are the
same. So, it is useful to have a way of changing the base of a logarithm.

KEY POINT 12.3

l loga x = logb x
logb a

where a ,b, x > 0

Proof 12.3

Prove that loga x = logb x .
logb a

Write the first statement in the Let log a x= y
equivalent form without logs Then x = ay

Then take log b of both sides So,
to introduce the new base
logbloxg= b a y

Use log xm = mlog x logb x y= a logb
b b
logb x
Make y the subject y = logb a

y = log a x That is,

loga x = logb x
logb a

WORKED EXAMPLE 12.8

Write log 5 in terms of natural logarithms.
3

Use log x = log x with log35 l=n5ln3
a b a

log
b
ln as the new logarithm

294 12 Analysis and approaches: Exponents and logarithms

WORKED EXAMPLE 12.9

Solve log3 x = logx81 .

Use loga x = logb x to change log3loxg=81 x
logb a
log x into log 3 so that the bases log 81
log3 x = 3
are the same on both sides
log3 x

log 3 81 = 4 log3 x = 4 x
log3

Multiply both sides by log 3 x (log3)x4 2 = −2
log3 x = ± 2
Square root both sides, x = 32or 3
remembering ± and
then solve for x

= 9 or 19

n Solving exponential equations

You have already met the technique of solving certain exponential equations by making
the bases the same on both sides of the equation. The only difference now is that you
might have to work with the new law of exponents introduced in Key Point 12.1.

WORKED EXAMPLE 12.10

( )Solve the equation 2 34 x − .x 1

=

For a Use a 1n = n a on the LHS (3)2 4x = x−1
reminder and write 4 as 22 on the RHS
of simple ( )1 x
exponential x−1
equations of this 23 = (2)2
type, see Section 1A.
x

232 = 2 x2−

Now equate the exponents So, x
and solve for x
3 = −2x2

x x= −6 6

5x6 =

x = 6
5

Similarly, you have also met exponential equations that needed logs to solve them.
Rather than being restricted to log 10 or ln, you could now be required to use log b for
any b > .0

12B Logarithms 295

For a WORKED EXAMPLE 12.11 5x7+3 =
reminder of
exponential Solve the equation 5 x+3 = 7 . x + 3=log 7 5
equations of this type
to base 10 or base e, Use a = bx is equivalent
see Section 1C. to log b a = x

Rearrange x l=og 7 3 −

5

Evaluate using your GDC = −1.79

When x appears in the exponent on both sides of the equation, but with different bases,
you may need to start by taking a logarithm of both sides. You can choose any base;
10 and e are the two most common choices.

WORKED EXAMPLE 12.12

Solve the equation 7x-2 = 5x+3 , giving your answer in terms of natural logarithms.

Tip Take natural logs of both sides ln (7)x−(2) = ln 5 x+3
(x) − 2 ln7 = (+x) 3 ln5
The final answer to Use aln m = mlan
xln7 2−ln7 ln5=3l+xn5
Worked Example 12.12 Expand the brackets x ln7 l−n5x3ln5=2l+n7

can be written in a Group the x terms and the
number terms (remember that
different form, by using ln 5 and ln 7 are just numbers)

laws of logarithms on Factorize the LHS and divide

the top and the bottom:

x= ln(537 ) 2 x(ln7 l−n5=3+l)n5 2ln7
ln(7 5÷)
x = 3ln5 2+ln7
ln(6125) ln7 l−n5
= ln(1.4) .

Exercise 12B

For questions 1 to 5, use the method demonstrated in Worked Example 12.4 to find the exact value of the logarithm.

1 a log 3 81 2 a log 1 ⎛⎞ 3 a log 64 8
2 ⎝8⎠

b log 4 16 b log 15 ⎛⎞ b log 16 2
⎝25⎠

4 a log 0.5 2 5 a log 2 1
b log 0.2 5 b log 7 1

296 12 Analysis and approaches: Exponents and logarithms

For questions 6 to 10, use the method demonstrated in Worked Example 12.5 to solve the equations.

6 a log3 x = −2 7 a log2 (y)− 1= − 3 8 a log52(x2)+ =
b log3 (y)− 2= − 2 b log43(x3)+ =
b log4 x = −2
10 a log4 (x)− 1=1
9a log9 z = 1
2

b log8 z = 1 b log5 (x)+ 3=0
3

For questions 11 to 14, use the method demonstrated in Worked Example 12.6. Write each given expression in terms of
p and q, where p = log 5a and q = log 5b.

11 a log 5⎛⎝⎜ab2 ⎞ 12 a log 5⎛⎝⎜ab32 ⎞ 13 a log 5 a 3b
⎠⎟ ⎠⎟

b log 5⎛⎝⎜ba3⎞⎠⎟ b log 5⎝⎛⎜ba42 ⎞ b log 5 a 4b3
⎠⎟

14 a log52⎛⎝5b2a ⎞
⎠

b log55⎛⎝⎜ba52 ⎞
⎠⎟

For questions 15 to 18, use the method demonstrated in Worked Example 12.7 to solve the equations, giving your

answers in an exact form.

15 a log10 x + log120 3 = 16 a log5 (x)+ 3+log 2 25 = 17 a log21(xlo)+g 2− 3 2 (x)− =

b log x + log 3 1= b log12 (x)− 1+log 4 112 = b log31(xlo)+g(−1)2 3 x − =

6 6

18 a ln(x)−(3−)ln x + 5=4

b ln(x)+(2−)ln x − 1=3

For questions 19 to 23, use the change of base formula from Key Point 12.3, with the method demonstrated in Worked
Example 12.8, to write each logarithm in the given base. Simplify your answer where possible.

19 a log 2 7 in terms of natural 20 a log 5 3 in base 2 21 a ln 4 in base 5
logarithms b log 7 5 in base 3 b ln7 in base 4

b log 5 8 in terms of natural
logarithms

22 a log 16 5 in base 2 23 a log 2 25 in base 5
b log 27 2 in base 3 b log 3 8 in base 2

For questions 24 to 26, use the method demonstrated in Worked Example 12.9 to solve the equations.

24 a log2 x = log x16 25 a 2log 5 x = log x25 26 a log29xlo=g 2 x
b log5 x = log x625 b 5log 2 x = log x32 b log41x6l=og 4 x

For questions 27 and 28, use the method demonstrated in Worked Example 12.10 to solve the equations.
x 2x
27 a ( )3 − 9x 1 ( )28 a 3 2 + 8x 1
= =

b ( )2 x + 4x 2 ( )b3 43 x − 9x 3

= =

For questions 29 to 31, use the method demonstrated in Worked Example 12.11 to solve the equations. Use your
calculator and give your answers to three significant figures.

29 a 3 17x = 30 a 5 x+1 = 11 31 a 2 113x1−=
b8 x= 3 b 7 x−1 = 12 b 3 4x3+= 8

For questions 32 and 33, use the method demonstrated in Worked Example 12.12 to solve the equations, giving your

answer in terms of natural logarithms.

32 a 3 x−2 = 2x+1 33 a 7 2x5− = 2 x+3

b 3 x−1 = 2x+2 b 7 3x1+ = 2x+8

12B Logarithms 297

34 Given that x = log 3 a, y = log 3b and z = log 3c, write the following in terms of x, y and z:

( )a log 3 ab4 b log 3⎛⎝⎜ac2b5 ⎞ ( )c log327 a 2b3
⎠⎟

35 Given that x = log5, a y = log5 b and z = log 5c, write the following in terms of x y, and z:

a log52(5 a) b log55⎝⎛ b⎞
c5⎠

36 Write 2ln a + 6ln b as a single logarithm.

37 Write 1 ln x − 1 ln y as a single logarithm.
3 2

38 a Find the exact value of log 1 2 ⎛ ⎞
⎝ 2⎠ .

b Solve the equation log x27 = −3.

39 Solve log2 (x)+ 3=3.

40 Solve the equation log32( x − 3=)4.

41 Use logarithms to solve these equations:

a 5 1x0= b 2 3× 6+2=x0
42 Solve the equation 3 1.1 x = 2. 0

43 Solve the equation log 3x2 5 = .

44 Solve the equation log 6x4 3 = .
45 a Given that 5 6× = × x, wr1it2e3dowx n the exact value of 2x .

b Hence find the exact value of x.
46 a Write log ex in terms of natural logarithms.

b Hence solve the equation 16ln x = log x e. .
47 Solve the equation 8 3x1+ = 4x−3.
48 Solve the equation 5 2x3+ = 9x−5, giving your answer in terms of natural logarithms.

49 The radioactivity (R) of a substance after a time t days is modelled by R 10 =0.9× t

a Find the initial (i.e. t = 0) radioactivity.

b Find the time taken for the radioactivity to fall to half of its original value.

50 The population of bacteria (B) at time t hours after being added to an agar dish is modelled by

B 1000=1.1 . t

a Find the number of bacteria

i initially

ii after 2 hours.

b Find an expression for the time it takes to reach 2000. Use technology to evaluate this expression.

51 The population of penguins (P) after t years on two islands is modelled by:
first island: P 2=00 1.1t
second island: P 1=00 1.2 . t

How many years are required before the population of penguins on both islands is equal?

52 Find the exact solution of the equation log x = 6log 8 .
53
4 x ln
Solve the equation 2 5−3 x = 32,x1− giving your answer in the form p
ln q , where p and q are integers.

54 Moore’s law states that the density of transistors on an integrated circuit doubles every 2 years. Write an

expression for the time taken for the density to multiply by 10 and use technology to evaluate this expression.

( )55 a If log a x2 = fbind the product of all possible values of x.
( )b If log a=xf2ind bthe product of all possible values of x.

298 12 Analysis and approaches: Exponents and logarithms

Checklist

n You should be able to extend the laws of exponents to general rational exponents:
a 1n = n a

n You should be able to work with logarithms to any (positive) base.

n You should know the laws of logarithms:

logb xy = logb x + logb y
x
logb y = logb x − logb y

logb xm = m logb x
where a ,x,y > 0

n You should be able to change the base of a logarithm:
logb x
loga x = logb a

where ,a,b x > 0

n You should be able to solve exponential equations.

n Mixed Practice 3
4⎞ −
9⎠ 2 .
1 a Find the exact value of ⎛⎝
b Find 1⎞
log2 ⎛ 8⎠ .
⎝

2 Write ln 4 +2ln 3 in the form ln k.

3 Solve the equation 2 3 x− 2 = 54.
4 Use technology to solve 1.05 x = 2.

5 Solve the equation 100x+1 = 103x .

6 Find the value of x such that log3(5 x1)+2= .

7 Find the exact solution of the equation 3ln x + 2= 2(ln 1)x − .

8 Given that =a log 3 x, =b log 3 y and =c log 3 z, write the following in terms of a, b and c:

( )a log 3 x2y b log 3⎛⎝⎜yxz3⎠⎞⎟ ( )c log 3 zx3

9 Find the value of each of the following, giving your answer as an integer.

a log 6 36 b log 6 4 + log 6 9 c log 6 2 - log 6 12

Mathematics SL May 2015 Paper 1 TZ1 Q2

10 Given thatx= log 2 a and =ylog 2 b, write in terms of x and y:
b log28⎝⎛⎜ab23⎞⎠⎟
a log 2⎛ a⎞
⎝ b⎠

11 Given thatx= ln 2 and =y ln5 , write the following in terms of x and y: c ln 0.08
a ln 10 b ln 50

12 Write3+2log5 2lo−g 2 as a single logarithm.
13 Find the exact solution of the equation 3ln x + ln8 5= , giving your answer in the form e A k where A

and k are fractions.

Mixed Practice 299

14 Solve the equation log3 x = 4log x3 .

15 Solve the equation 43x5+ = 8x−.1

16 Solve the simultaneous equations:

log2 x + log3 y = 5

log2 x − 2log 3 y =− 1

17 Find the exact value of x such that logx8 6= .

18 Solve the equation 32x = 2ex , giving your answer in terms of natural logarithms.

19 Solve the equation 52x1+ = 7x−.3 x i1n the form log p, where p and q are positive integers.
log q
20 Find the solution of the equation 122x = ×4+3

21 The number of cells in a laboratory experiment satisfies the equation = 1N50e1.04 t , where t hours is
the time since the start of the experiment.

a What was the initial number of cells?
b How many cells will there be after 3 hours?

c How long will it take for the number of cells to reach 1000?

22 A geometric sequence has first term 15 and common ratio 1.2. The nth term of the sequence equals
231 to the nearest integer. Find the value of n.

23 Find the value of b 8 log 2 5
a log240 l−og 5 2 ;

Mathematics SL May 2013 Paper 1 TZ1 Q7

24 a Find log 2 32. ⎝⎛⎜382yx⎠⎞⎟can
b Given that log2
be written as px + qy, find the value of p and of q.

Mathematics SL May 2009 Paper 1 TZ2 Q4

25 Let log3p = 6 and log3q = 7. b Find log 3⎛⎝⎜qp⎠⎞⎟. c Find log (93 ) p .
a Find log 3 p2.
Mathematics SL May 2013 Paper 1 TZ2 Q3

26 Solve the equation 8x−1 = 63x. Express your answer in terms of ln2 and ln3.

Mathematics HL May 2014 Paper 1 TZ2 Q2

27 Solve the equation 3x+1 = 3+x18 .

28 Given that =x ln a and =y ln b, express log ba in terms of x and y.

29 Given that =x log b a, express the following in terms of x:
b log 2a b .
a logb ⎛a2 ⎞
⎝⎜b3 ⎠⎟

30 Write in the form k ln x, where k is an integer:

ln x + ln x2 + ln x3 + + ln x20

31 Find the exact value of

log3 ⎛ 1⎞ + log3 ⎛ 3⎞ + log3 ⎛ 5⎞ + + log3 ⎛ 79⎞
⎝ 3⎠ ⎝ 5⎠ ⎝ 7⎠ ⎝ 81⎠

Analysis and approaches:

13 Sequences and series

ESSENTIAL UNDERSTANDINGS

n Algebra is an abstraction of numerical concepts and employs variables to solve mathematical
problems.

In this chapter you will learn...

n how to find the sum of infinite convergent geometric sequences
n how to use the binomial theorem to expand expressions of the form(a)nb+ , where n ∈ +
n how to use Pascal’s triangle to find the coefficients in a binomial expansion
n the formula for nC r, which can be used to find the coefficients in a binomial expansion.

CONCEPTS

The following key concepts will be addressed in this chapter:
n The binomial theorem is a generalization which provides an efficient method for

expanding binomial expressions.

PRIOR KNOWLEDGE

Before starting this chapter, you should already be able to complete the following:
1 Find the sum of the first 20 terms of the geometric sequence 80, 20, 5, 1.25, …
2 Use technology to find the limit of the function f(x) = 0.5x as x gets larger.
3 Find the range of values of x for which | x | < 1.
4 Expand and simplify (3 - 4x)2 .

n Figure 13.1 How do mathematicians reconcile the fact that some conclusions conflict with intuition?

Analysis and approaches: Sequences and series 301

It might seem natural that if you continue adding terms that are all positive in a
sequence the sum will just get larger and larger, and that if you continue adding terms
that are all negative the sum will just get more and more negative. This is certainly the
case with many sequences. It is also possible in some sequences, however, that no matter
how many positive (or negative) terms you add, the sum can never exceed a certain
value. In particular, this will happen with certain geometric sequences.

Starter Activity

Figure 13.1 shows the first four stages of building a Koch snowflake. At each stage the
external sides are split into three and an equilateral triangle is added to the middle. What
happens to the perimeter of this shape as the stages progress? What happens to the area?

Now try this problem:

Using technology, investigate the value, S, of the infinite geometric series
S = 1+ +r +r r+2… 3

when
a r 2=
b r 1=
c r 0=.25
d r = −0.25
e r = −1
f r = −2.

302 13 Analysis and approaches: Sequences and series

13A The sum of infinite convergent

geometric sequences

You met You know that you can find the sum of the first n terms of a geometric sequence with
the sum of n
a geometric u1r(1 −)
sequence in Section common ratio r and first term u 1 using the formula Sn = 1− r . Sometimes this sum
2B.
will just increase (or decrease, if negative) the more terms you add.
You met
the idea However, if r is between 1 and -1 then as n gets very large, rn tends towards 0.
of limits in u1(1 − r n ) u1(1 −0) = 1−u1r .
Section 9A. As a result, 1− r tends towards 1− r

Tip So, in this situation the sum of infinitely many terms converges to a finite limit – this is

Remember that | r | < 1 called the sum to infinity, S ∞, of the geometric sequence.
is just another way of
KEY POINT 13.1
writing r is between -1
For a geometric sequence with common ratio r,
and 1. 1−u1 r
S = if | r | < 1

∞

WORKED EXAMPLE 13.1

Find the value of the infinite geometric series + +2 4 8 16
+ 3+… 9 27

This is a geometric series with u1 = 2=, r 2
2 3
first term 2 and common ratio 3

Use S∞ = 1−u1 r S = 1 2
−2
∞

3

=6

42 TOOLKIT: Proof

∞ Does the following diagram prove that

3 1 + 1+16… =614 13?
4
π924

13A The sum of infinite convergent geometric sequences 303

You are the Researcher

The ancient Greeks had some real difficulties working with limits of series. Some of them thought,

incorrectly, that if each term in the sum got smaller and smaller, then the series would reach a limit. Howev
s2er+ie13+s,…114and
this is not always true. You might like to explore the harmonic see why that does not
converge.

There are many tests that mathematicians use to decide if a series will converge, for example, the ratio test
You might like to learn more about these and how they work.

WORKED EXAMPLE 13.2 converges.

The geometric series ( )x +( 4+) (+ )+x + 4+…2 x 4 3 r= x+ 4
Since the series converges,
Find the range of possible values of x. | x + 4| < 1
State the common ratio −1< +x< 4 1
−5< <x− 3
You know that | r | < 1

Be the Examiner 13.1

Find the sum to infinity of the geometric series,
1 +43 9 27
2 − − 8 16 +…

Which is the correct solution? Identify the errors made in the incorrect solutions.

Solution 1 Solution 2 Solution 3
r = − 34÷ =12− so i23t
S∞ = 1 1 doesn’t exist.
1 −1 2
( )S∞= 1− 2 3
=2 − 2

= 1
5

42 TOOLKIT: Problem Solving

∞ You may not have realized it, but you have already met infinite geometric series in
your previous work. Explain how you can find the following sum using methods
3 from your prior learning. Confirm the answer by using the formula for the geometric
series.
π924
∑∞ 3
10r

r =1

What happens to your argument when applied to the sum below?

∑∞ 9
10r

r =1

304 13 Analysis and approaches: Sequences and series

42 TOOLKIT: Proof

∞ Intuition for sums tends to break down when dealing with infinite sums.
For example, consider
3
S 1= 1−1+1−1 +1 − …
π924 Consider the following attempts to evaluate S:

Attempt A:

S = (1 - 1) + (1 - 1) + (1 - 1) …

= 0+ 0+ 0…

=0

Attempt B:

S = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) …

= 1+ 0+ 0+ 0…

=1

Attempt C:

S= 1- 1+ 1- 1+ 1- 1…

S = +1 - 1 + 1 - 1 + 1 - 1 …

Adding these together gives:

2S = 1 + 0 + 0 + 0 …

S= 1
2

All of these look plausible, but none are considered correct because you cannot
generally group or rearrange terms in an infinite sum.

Exercise 13A

For questions 1 to 5, use the method demonstrated in Worked Example 13.1 to find the sum of the infinite geometric series.

1a 1 + 1+16 1 + … 2a 1 + 92+ + 2…47 3 a 4 +1 + +41…
4 64 3

b 1 + 41+ +18… b 12 4 b 6 +2 + +32…
2 3 + 1+5+ …75

4a 1 − +14 − 18+… 1 5a 15 −9 + −25+7… 81
b 2 16 b 25

1 − 19+ 1 − 8+11… 16 −12+9− +… 27
3 27 4

For questions 6 to 10, use the method demonstrated in Worked Example 13.2 to find the range of values of x for which
the infinite geometric series converges.

6 a (x − 2+) −( x 2)2(+ −x 2)3 +… 7 a + +1 2 4x2 x + … 8a 1 +2 4+x x+ 2

…

b ( x + 3+) +( x 3)2(+ +x 3)3 +… b +1+3 9x2 x + … b +1+5 +2x5x… 2

9 a (x + 4−) +( x 4)2(+ +x 4)3 −… 10 a − +1 3x 9 x2
b ( x1)−(−1)−( +x1)− 2 x 3 −… 2 4 −…
16 x2
b −1+ 4x 9 −…
3

13A The sum of infinite convergent geometric sequences 305

11 A geometric series has first term 3 and common ratio 1 4. Find the sum to infinity of the series.
12 and common ratio − 41 .
13 Find the sum to infinity of the geometric series with first term 5

Find the sum to infinity of the series − 2+ 2 2 2
−3+ …9 27

14 An infinite geometric series has the first term 8 and its sum to infinity is 6. Find the common ratio of the series.

15 The first term of a geometric series is 3. Given that the sum to infinity of the series is 4, find the value of the

common ratio. 1 . Find the first three terms of the series.
16 The sum to infinity of a geometric series is 3 and the common ratio is 3

17 The second term of a geometric series is 2 and the sum to infinity is 9. Find two possible values of the common

ratio. ∞2 r

∑18 ⎛⎞ .
⎝5⎠
A geometric series is given by r =0

a Write down the first three terms of the series.

b Find the sum of the series.
∞2
Evaluate r =0 3r .
a Find the range of
∑19 values of x for which the geometric series − +3−3 +2x…7x2x423 3 converges.

20

b Find the sum of the series when x = - 2.
21 A geometric series is given by +5 5( x − 3+) 5( 3)x2− + …

a Find the range of values of x for which the series converges.

b Find the expression, in terms of x, for the sum of the series.
22 For the geometric series +2 4 x + 8x2 + …

a Find the range of values of x for which the series converges.

23 b Find the expression, in terms of x, for the sum to infinity of the series. the first term of the series.
The second term of a geometric series is − 56and the sum to infinity is 5. Find

24 Given that x is a positive number,

a Find the range of values of x for which the geometric series + x+x 4 136 x5 + … converges.

b Find an expression, in terms of x, for the sum to infinity of the series.
∞ xr +1
3.
∑25 a Find the value of x such that r =0 = 2r
b Find the range of values of x for which the series converges.

26 An infinite geometric series has sum to infinity of 27 and sum of the first three terms equal to 19. Find the first term.

306 13 Analysis and approaches: Sequences and series

13B The binomial expansion

n The binomial theorem

If you expand (a + b)2 , (a + b)3 , (a + b)4 and so on, by multiplying out brackets
repeatedly, you will get:

(a b+ )2 = a 2 + 2ab b+ 2
(a b+ )3 = +a3 3a 2b a+b3b 2 3+
(a b+ )4 = a4 + 4a3b + 6a2b2 + 4ab3b4+

You can see that there are patterns in the powers of a and b in each expansion – the
power of a decreases by one as the power of b increases by one between terms.

There are also symmetrical coefficients (in red) – these are often referred to as the
binomial coefficients. The coefficient in the term containing br of the expansion of
(a + b)n is called n C r. They can be found using the n C r button on your calculator. For
example, the coefficients in the expansion of (a + b)4 are found using:

4 C01=C 4 4C 61 C= 4 C4 1 2 = 4 3 = 4 4 =

Tip

Note that since 4C 0 = 1 and 4C 4 = 1, these coefficients do not need to be written in the
expansion (this is the same in the first and last terms of all of these expansions).

Tip KEY POINT 13.2 + … + bn

Remember that n ∈ + (a b+ )n = a n n+ C1a nb−1 +… + nCr a nb−r r
means that n is a where n ∈ +.
positive integer.

13B The binomial expansion 307

WORKED EXAMPLE 13.3

Find the first three terms in increasing powers of x in the expansion of (2 + x)5 .

Use (2 )+2x=C5+525 1 4 x1 5 +2 C 2 3 x2 + …
(a b+ )n = a n n+ C1 a nb−1 + nCn 2 a b−2 2 + …
= 3+2 5(16)( ) 1x0(+8)( ) x2 + …
Evaluate the coefficients and simplify = 3+2+80+…80x x 2

CONCEPTS – GENERALIZATION

If you had tried to multiply out (2 + x)5 manually you would have seen that the binomial
expansion is a much more efficient way to get to the same result. You might have
thought that with technology such as algebraic computer systems we no longer really
need to study these algebraic generalizations; however, there are many reasons
why we still do so. In many technological situations millions of algebraic calculations
are required – for example, when modelling the air flowing around an aeroplane or
animating droplets of water in a cartoon. In these situations, using the most efficient
algebraic generalizations can massively improve performance, as the computer does not
need to go back to first principles in every calculation.

Tip WORKED EXAMPLE 13.4

Make sure to read Find the coefficient of y3 in the expansion of (2 - 3y)7 .
carefully whether the
question wants the Use the general term, nCr anb−r r , The relevant term is
coefficient, –15 120, or with =n 7, r = 3, a = 2, b = −3 7 C3 2 4( 3−) y 3
the term, -15 120y3 .
= (35)(16)(−27 )y3
You are asked for the coefficient = −15 120y3
(the number in front of y3 )
So, the coefficient of y3 is -15 120

Be the Examiner 13.2

Expand and simplify (5 - 4x)3 .
Which is the correct solution? Identify the errors made in the incorrect solutions.

Solution 1 Solution 2 Solution 3

(5 −4 ) x 3 = 5+3 3C152( 4− ) x (5 −4 ) 5x 3 = +3 3 C152( 4− )x x3 (5 −4 )=5x+C3 353 ( 4 ) 1 2 − x
+ 3C25 1( 4− ) x( 42 )+ − x 3 + 3C25 1( 4− )x( 42 )+ − + 3C251(24− )+x( −4 ) x 3
3
= 125 −100 x + 80x2 − 64x3 = 125 −300 60x 4− x2x− = 1−2+5 300 24x0x6x4 2 3−

308 13 Analysis and approaches: Sequences and series

n Use of Pascal’s triangle and the factorial formula

to find n C r

As well as getting the binomial coefficients from the n C r button on your calculator, you
can also get them from Pascal’s triangle, which is formed by adding pairs of adjacent
numbers to give the number between them in the row below:

n=0 1
n=1 11
n=2 121
n=3 1331

n=4 14641

n=5 1 5 10 10 51

This enables you to do binomial expansions for small values of n without your GDC.

WORKED EXAMPLE 13.5

Expand and simplify (1 - 2x)4 .

Use (a b+ )n n=na+n C1a b−1 (1 −2 ) x1 C4 4=14 1(+2 ) 3 1− x x4
+ nC2a nb−2 2 +… + 4C2 12(22− ) x
+ 4C3 11(3−2 )+x(−2 )
Use the row starting 1, 4, ... from
Pascal’s triangle to evaluate = 1+4−(1+)( 2 ) 6x(1)(4 ) x2
the binomial coefficients: + 4−(1+)( 8 )x31x6
1 4
11 = 1−8+2−4x 3x+2x1x62 3 4
121
1331
14641

Although in the Western world this famous pattern is attributed to the French
mathematician Blaise Pascal (1623–1662), there is evidence that it was known
much earlier in Chinese work, such as that of Yang Hui (1238–1298).

There is also a formula for the binomial coefficients. This involves the factorial
function:

n!n=n× − ×…( × 1× ) 21

Note that 0! is defined to be 1.

13B The binomial expansion 309

KEY POINT 13.3

● nC = n! )!
r r !n( r −

Tip

In some other resources, and some past paper questions, you might see n C r referred to using the
n

( )r .

notation

This formula can be proved in various ways. If you study Mathematics: Analysis and
approaches HL you will learn about ways to prove it using induction (Chapter 5) and
consider it as a way of choosing options (Chapter 1).

WORKED EXAMPLE 13.6

Evaluate 12C 10.

Use nC = n! )! 12 C = 12!
r r !n( r − 10 10!(12 1−0)!

= 12!
10!2!

12! =12× ×1× 1…× 1× 0 9 Note that = 1× × 2 11 10!
21 10!2!

Cancel 10! and evaluate = 12 11
2

= 6× 11

= 66

WORKED EXAMPLE 13.7

Using technology, find the possible values of x when 10C x = 210.

Use the table function to From GDC,
generate values of 10C x for x = 4 or 6

different values of x

310 13 Analysis and approaches: Sequences and series

42 TOOLKIT: Problem solving

∞ There are many identities associated with binomial coefficients. Investigate and see if
you can justify expressions for:
3
n−1Cr + n−1Cr −1
π924
∑ n n Cr
r =0

∑ ( )n nCr 2
r =0

You are the Researcher

You might want to investigate how the binomial expansion generalizes to more
complicated expansions, as described by the multinomial theorem.

Exercise 13B

For questions 1 to 8, use the method demonstrated in Worked Example 13.3 to find the first four terms of the binomial
expansion in increasing powers of x or b.

1 a (2 + x)6 2 a (3 - x)5 3 a (1 + 2x)6
b (2 + x)7 b (3 - x)4 b (1 + 2x)7

4 a (1 - 5x)4 5 a (2 + 3x)10 6 a (2 - 3x)5
b (1 - 5x)5 b (2 + 3x)9 b (2 - 3x)6

7 a (a + 2b)7 8 a (3a - 2b)5

b (a + 2b)8 b (3a - 2b)6

For questions 9 to 14, use the method demonstrated in Worked Example 13.4 to find the required coefficient.

9 a y4 in (2 + y)8 10 a y5 in (3 - y)9 11 a x6 in (1 - 2x)10
b y4 in (2 + y)9 b y5 in (3 - y)10 b x6 in (1 - 2x)9

12 a x5 in (2 + 3x)12 13 a z2 in (2z - 1)20 14 a z7 in (2z - 3)12

b x5 in (2 + 3x)11 b z2 in (2z - 1)19 b z7 in (2z - 3)11

For questions 15 to 17, use Pascal’s triangle from Worked Example 13.5 to expand the brackets.

15 a (1 + 2x)4 16 a (3 + x)3 17 a (x + 2)5

b (2 + x)4 b (1 + 3x)3 b (x + 1)5

For questions 18 to 22, use the method demonstrated in Worked Example 13.6 to evaluate these binomial coefficients:

18 a 6C 1 19 a 7C 2 20 a 9C 3

b 8C 1 b 6C 2 b 8C 3

21 a 10C 9 22 a 15C 13

b 13C 12 b 11C 9

For questions 23 to 25, use technology, as in Worked Example 13.7, to find the unknown value (or values).

23 a 14C r = 2002 24 a 20C r = 1140 25 a n C 5 = 792
b n C 8 = 43 758
b 15C r = 6435 b 17C r = 6188

26 Expand and simplify (10 - 3x)4 .

27 Expand and simplify (2x - 1)5 .

28 Find the coefficient of x5 in the expansion of (5 + 3x)7 .

29 Find the coefficient of x8 in the expansion of (3x - 2)12 .

30 Find the value of n such that n C 4 = 495.

Mixed Practice 311

31 a Find the first three terms in the expansion of (2 + 3x)10 .
b Hence find an approximate value of 2.00310 , correct to the nearest integer.

32 a Find the first four terms, in ascending powers of x, in the expansion of (3 - 2x)9 .
b Hence find the value of 2.989 correct to one decimal place.

33 a Find the first three terms, in ascending power of x, in the expansion of (2 + x)7 .

b Hence expand (2 + x)7 (3 - x) up to and including the term in x2 .
34 a Find the first three terms in the expansion of (2 - 3x)5 .

b Hence find the coefficient of x2 in the expansion of (2 - 3x)5 (1 + 2x).
35 Find the coefficient of x2 in the expansion of (2 - x)5 (1 + 2x)6 .
36 a Expand and simplify (2 + x)4 - (2 - x)4 .

b Hence find the exact value of 2.014 - 1.994 .
37 The coefficient of xn - 3 in the expansion of (x + 2)n is 1760. Find the value of n.
38 The coefficient of x4 in the expansion of (x + 3)n is 153 090. Find the value of n.

( )39 Expand and simplify x x + 2. 4

40 Expand and simplify (x2 + 3x)5 .

41 Find the coefficient of x27 in the expansion of (x2 + 3x)15 .

( )42 2 2 3 10
Find the coefficient of x5 in the expansion of x − x
.

43 Find the coefficient of x9 in (x - 1)7 (x + 1)7 .

Checklist

n You should be able to find the sum of infinite geometric sequences:

For a geometric sequence with common ratio r,

S = u 1 if | r | < 1
1−
∞ r

n You should be able to use the binomial theorem to expand expressions of the form (a + b)n , where n ∈ +:
(a b+ )n = a n + nC1a nb−1 +… + nCr a nb−r r + … + bn

n You should be able to use Pascal’s triangle to find the coefficients in a binomial expansion.

n You should be able to use the formula for n C r to find the coefficients in a binomial expansion:
n!
nC = r !n( r − )!

r

n Mixed Practice

1 Expand and simplify (2 + x)4 .
2 Find the sum of the infinite geometric series 13, 91, 217 ,…

3 A geometric series has first term 7 and common ratio − 5ra9ti.oFind43 the sum to infinity of the series.
4 Find the first term of the geometric series with common and sum to infinity 12.

5 The coefficient of x6 in the expansion of (x + 1)n is 3003. Find the value of n.

312 13 Analysis and approaches: Sequences and series

6 In the expansion of (3x - 2)12 , the term in x5 can be expressed 1a2s Cr (3 ) (x2)p × − . q
a Write down the value of p, of q and of r.
b Find the coefficient of the term in x5 .

Mathematics SL May 2013 Paper 2 TZ1 Q3

7 Consider the expansion of (x + 3)10 .
a Write down the number of terms in this expansion.
b Find the term containing x3 .

Mathematics SL May 2014 Paper 2 TZ1 Q2

8 a Find the first three terms in the expansion of (1 + 2x)10 .
b Use your expansion to find an approximate value of 1.00210 .

9 An infinite geometric series is given by
(2 3− ) x(2 +3 )−(2 3 x) 2 + − x 3 +…
a Find the range of values of x for which the series converges.
b Given that the sum of the series is 21, find the value of x.
c Show that the sum of the series cannot equal − 2 3.

10 The sum to infinity of a geometric series is three times larger than the first term. Find the common
ratio of the series.

11 Find the coefficient of x4 in the expansion of (2 - x)10 .
12 Given that(+3 ) x n = 8+1+…kx

a Find the value of n.
b Find the value of k.

13 a Write down an expression, in terms of n, for n C 1.
b Given that ( x + 2)n = xn + 18x1n− + bxn−2 + …
i Find the value of n.
ii Find the value of b.

14 The third term in the expansion of (2x + p)6 is 60x4 . Find the possible values of p.

Mathematics SL November 2012 Paper 2 Q4

15 The sum of the first three terms of a geometric sequence is 62.755, and the sum of the infinite
sequence is 440. Find the common ratio.

Mathematics SL May 2013 Paper 2 TZ2 Q5

16 Evaluate

∑∞ 3r4+ r
5r

r =0

17 a Explain why the geometric series e− x + e−2x + e−3x + … converges for all positive values of x.
b Find an expression for the sum to infinity of the series.
c Given that the sum to infinity of the series is 2, find the exact value of x.

Mixed Practice 313

18 A geometric series has sum to infinity 27, and the sum from (and including) the fourth term to
infinity is 1. Find the common ratio of the series.

that⎝⎛1 2 x⎞ n
3 ⎠
19 Given + (3 + nx)2 = 9+ 84 x +… fi,nd the value of n.

Mathematics SL May 2012 Paper 1 TZ2 Q7

⎛. x⎞ 6
⎝ ⎠
20 Find the coefficient of x-2 in the expansion of (−x1) 1 23 x +

Mathematics HL May 2014 Paper 2 TZ2 Q5

21 Let {u n}, n∈ +, be an arithmetic sequence with first term equal to a and common difference of d,
where d ≠ 0. Let another sequence {v n}, n∈ +, be defined by v n = 2u
n.

a i Show that d+ ownvvntnh1eifsirastcotenrsmtanotf. the
ii Write
sequence {v n}.

iii Write down a formula for v n in terms of a, d and n.

Let S n be the sum of the first n terms of the sequence {v n}.

i Find S n , in terms of a, d and n.
∑ ∑b v
ii Find the values of d for which ∞ i exists.
You are now told that i=1
∞v i
does exist and is denoted by S ∞.
i=1
iii Write down S ∞ in terms of a and d.

iv Given that S ∞ = 2a + 1 find the value of d.

Let {w n}, n ∈ +, be a geometric sequence with first term equal to p and common ratio q, where p and
q are both greater than zero. Let another sequence {z n} be defined by z n = lnw n.

∑c Find n giving your answer in the form lnk with k in terms of n, p and q.

i=1z i Mathematics HL May 2015 Paper 1 TZ1 Q12

Analysis and approaches:

14 Functions

ESSENTIAL UNDERSTANDINGS

■ Creating different representations of functions to model relationships between variables, visually
and symbolically, as graphs, equations and tables represents different ways to communicate
mathematical ideas.

In this chapter you will learn...

■ how to form composite functions
■ when composite functions exist
■ what the identity function is
■ how to find inverse functions
■ when inverse functions exist.

CONCEPTS

The following key concepts will be addressed in this chapter:
■ Functions represent mappings that assign to each value of the independent variable

(input) one and only one dependent variable (output).

PRIOR KNOWLEDGE

Before starting this chapter, you should already be able to complete the following:

1 If f(x) 3=4−, calxculate − f( 2).
2 Find the largest possible domain of f(x) = +x 1 3.

3 a Find the range of f(x) x= +2+ ∈4x1,x .

b State whether xf( ) is a one-to-one or many-to-one function.

■ Figure 14.1 Does the order in which we do things always matter?

Analysis and approaches: Functions y 315
y = f(x) x
4 Sketch the inverse function of the
following graph:

5 Make x the subject of the following:
3x1+
ay = x−1

b y e= 2−x

Functions can be combined in many different ways: they can be added, multiplied,
divided or we can take one function and apply it to another function. Sometimes the
order in which functions are combined does not matter, but at other times we get
different results depending on the order in which the functions have been combined.

Starter Activity

Look at the pictures in Figure 14.1. In small groups label each part of the process of baking
a cake shown. Does the order in which they occur matter? Can each process be reversed?
Now look at this problem:
Ann thinks of a number and squares it. She then adds five to the result.
Bill thinks of a number and adds five to it. He then squares the result.
Both Ann and Bill end up with the same answer.
a What was the answer?
b Will Ann and Bill always end up with the same answer?

316 14 Analysis and approaches: Functions

Tip 14A Composite functions

You must apply the If, after applying one function, g, to a number you then apply a second function, f, to
function nearest to x the result, you have a composite function.
first. So, f(g(x)) or
(f g)(x) means you This is written as f(g(x)) or (f g)(x) or simply fg(x).
apply g first and then f.
WORKED EXAMPLE 14.1

f(x) 2=3−axnd x g( ) = x2
Find

a f(g(x)) b g(f(x)).

Replace x in f(x) with g(x) a f(g(x)) f=( ) x2
Replace x in g(x) with f(x) = 2−x32

b g(f(x)) g=(2 3) x −
= (−2 x3)2

Be the Examiner 14.1

f(x) x= + 4 and g(x) 5= x
Find (f g)(3).

Which is the correct solution? Identify the errors made in the incorrect solutions.

Solution 1 Solution 2 Solution 3

f(3) =3 4+7= (f g)(3) 5(=3 4+) (f g)(3) 5 3= 4× +
= 5× 7 = 1+5 4
g(3) 5= 3× =15 = 35 = 19
So, (f g)(3) 7 1=5× =105

42 TOOLKIT: Problem Solving

∞ If f(g(x)) g≡(f( )) xthen the two functions are said to be commutative.

3 Find examples of two functions f(x) and g(x) which commute. Can you make and
prove any general conjectures about some types of functions which always commute
π924 with each other?

Can you find a function f(x) which commutes with any other function?

For a composite function f(g(x)) to exist, all possible outputs from g must be allowed as
inputs to f, i.e. the range of g must lie entirely within the domain of f.

14A Composite functions 317

You met WORKED EXAMPLE 14.2
domain and
range in f(x) l=n( 2) x, x+ > −2
Section 3A. g(x) x= + 1

Find the largest possible domain of (f g)(x).

The only thing restricting the Largest possible domain of (f g)
domain is that the range of g(x) occurs when g(x) 2> −
must fit into the domain of f(x)
x +1 2> −
The domain of f is >x − 2. x > −3
So, the largest the range
of g can be is g(x) 2> − This is the largest possible domain.
Solve the inequality for x

Exercise 14A

For questions 1 to 8, use the method demonstrated in Worked Example 14.1 to find an expression for the composite
function (f g)(x).

1 a f(x) = 3x - 1, g(x) = x2 2 a f(x) = x2 ,g(x) 2=1+x 3 a f(x) x3=x2+ 2 , g(x) x2=
b f(x) = 4x + 2, g(x) = x2
b f(x) = x2 g, (x) 3=2−x b f(x) x5=x3− 2 , g(x) = 2x
4 a f(x) = 3ex , g(x) = 2x + 5 6 a f(x) x=x−3 2 , g(x) e=x
5 a f(x) 3=1+, x g(x) 4=e x
b f(x) = 4ex , g(x) = 3x + 1 b f(x) x=x+3 4 , g(x) e=x
7 a f(x) = +x 1 1, g(x) = 3x + 2 b f(x) 2=5+, x g(x) 3=e x

b f(x) = −x 1 2, g(x) = 2x + 5 8 a f(x) 3=2−2x,x g(x) = 1
x

b f(x) 4=3+2x, x g(x) = 1
x

For questions 9 to 14, use the method demonstrated in Worked Example 14.2 to find the largest possible domain for the
function (f g)(x). Where the domain of f is not given, assume the largest possible real domain.

9 a f( x) ln=( x + 3, )x > -3 and g(x) = x + 5 10 a f( x) ln=(4−), x < x4 and g(x) = x + 3

b f( x) ln=( x + 8,)x > -8 and g(x) = x + 1 b f( x) ln=(1−), x <x1 and g(x) = x + 1

11 a f( x) = 2x1+ , x 13−a21ndangd(xg)(x=) = 3x - 2 12 a f( x) 2=1 x − and g(x) = 4 - x
b f( x) = x 2x - 3
f( x) = 3x1+ , ≠2 b f( x) 3=1− axnd g(x) = 2 - x
13 a −x 1 2, x and g(x) = 4x + 1 14 a f( x) = −x 1 3 and g(x) = ex

b f( x) = +x 1 5, x ≠ -5 and g(x) = 2x - 7 b f( x) = −x 1 7 and g(x) = ex

15 Let f(x) = 3x - 1 and g(x) = 4 - 3x. f)(x).
a Find and simplify and expression for (f g)(x).

b Solve the equation (g f)(x) = 4.
16 Let f(x) = x2 + 1 and g(x) = x - 1.

a Find and simplify an expression for f(f(x)).

b Solve the equation f(g(x)) = g(f(x)).
17 Given that f(x) = 2x3 , find a simplified expression for (f

318 14 Analysis and approaches: Functions

18 Given that f(x) = x − 4, x 4 and g(x) = 3x + 10,

a Find the largest possible domain for the function (f g).

b Solve the equation (f g)(x) = 5. f)(8).
19 Let f(x) = lnx(x > 0) and g(x) = x - 5.

a Write down the exact values of (f g)(8) and (g

b Solve the equation fg(x) = 8.
20 Functions f and g both have domain {1, 2, 3, 4, 5} and their values are given in the following table.

x 12345
f(x) 5 4 3 2 1
g(x) 3 1 4 5 2

a Find

i (f g)(3) ii (g f)(4)

b Solve the equation (f g)(x) = 1
21 Some of the values of the functions f and g are given in the following table.

x 13579
f(x) 3 7 5 1 9
g(x) 5 7 9 3 1

a Find

i (f g)(3) ii (g f)(9)

b Solve the equation (f g)(x) = 5

22 Let f(x) = lnx, x > 0) and g(x) = x - 3.
a Find the largest possible domain for the function (f g).

b Solve the equation (f g)(x) = 1.

c Solve the equation (g f)(x) = 1.

d Solve the equation (f g)(x) = (g f)(x).

23 Given that f(x) =x + 1 2 and g(x) = −x 1 ,
3

a Find and simplify an expression for (f g)(x).

b Find the largest possible domain for the function (f g)(x).

c Solve the equation (f g)(x) = 2. for some constant k, which you should find.
24 Let f(x) =x 4 , x ≠ 0and g( x) 2=, x2x ∈ . Prove that gf (x) kf≡gx( )

25 Let f(x) = 1 for x ≠ − 23.
3x2+

a Find the largest possible domain for (f f)(x).

b Use technology to find the range of (f f)(x) for the domain from part a.

c Solve the equation (f f)(x) = 1.

14B Inverse functions 319

14B Inverse functions

■ Identity function

You have already met the idea that the inverse, f -1 , of a function f reverses the effect
of f. This can be expressed more formally in terms of composite functions.

KEY POINT 14.1

(f f ) (−)1 (xf = −1 f )(x) x=

A function that has no effect on any value in its domain is called the identity function.
So, the functions (f f) −1 and (f −f1) are both the identity function.

■ Finding the inverse function

In order to find an expression for f -1 from f you need to rearrange f to find the input (x)
in terms of the output (y).

KEY POINT 14.2

To find an expression for f -1(x):

1 Let y = f(x).

2 Rearrange to make x the subject.
3 State f -1(x) by replacing any ys with xs.

WORKED EXAMPLE 14.3

f(x) = 3x 5−
2

Find the inverse function, f -1.

Let y = f(x) y = 3−x52
Rearrange to make x the subject
2 y3 x=5 −

2 y5 +3 = x

x = 2 y5+
3

Write the resulting So, f −(1) x2 5= x+
function in terms of x 3

320 14 Analysis and approaches: Functions

See Section ■ The existence of an inverse for one-to-one functions
3A for a
reminder For f -1(x) to be a function, it must map each input value to a single output value. But
of one-to-one since the graph y = f -1(x) is a reflection in y = x of the graph y = f(x), it follows that for
and many-to-one f(x) each output must come from a single input, i.e. f(x) must be one-to-one.
functions. If f(x) is many-to-one, f -1(x) would not be a function as there would be input values that
map to more than one output value.

KEY POINT 14.3

A function has to be one-to-one to have an inverse.

Any function can be made one-to-one (and therefore to have an inverse) by restricting
its domain.

WORKED EXAMPLE 14.4

Use technology to find the largest possible domain of the function f(x) = xln x of the form
x kfor which the inverse f -1 exists.

f-1 will only exist if f is one-to- Largest possible domain of
one. Use the GDC to sketch the f for which f -1 exists:

graph and find the minimum x 0.368
point. Eliminating values of
x to the left of the minimum
will leave f being one-to-one

Be the Examiner 14.2

Find the inverse function of f(x) = x2 - 4, x < 0.
Which is the correct solution? Identify the errors made in the incorrect solutions.

Solution 1 Solution 2 Solution 3

y x= 2 − 4 y x= −2 4 f isn’t one-to-one,
x2 = +y 4 x2y=4+ so the inverse
x = y +4 x = − +y 4 function doesn’t
So, f -1(x) = x +4 So, f -1(x) = − +x 4 exist.

14B Inverse functions 321

Exercise 14B

For questions 1 to 10, use the method demonstrated in Worked Example 14.3 to find an expression for the inverse

function, f -1(x). x+
2
1a f( x) 4=1 2 a f(x) = 4x - 3 3 a f(x) = e4x 4 a f(x) = 3ex - 2
b f(x) = 5x + 1 b f(x) = e3x b f(x) = 2ex + 3
b f( x) 3=4 x+
5

5a f( x) lo=g (3 1) x+ 6 a f( x) = x − 2 7 a f(x) = (x + 2)3 8 a f(x) = x3 - 2
b f( x) = x + 3 b f(x) = (x - 3)3 b f(x) = x3 + 5
2

b f( x) lo=g (4 13) x −

9a f( x) = +xx − 3 10 a f( x) 2=1 x+
b f( x) = +xx − 2 3x2−
1 x
3 b f( x) 3=1 2x − 3
−

For questions 11 to 17, use the method demonstrated in Worked Example 14.4. For each function f find the largest

possible domain of the given form such that f has an inverse function.

11 a f(x) = (x - 2)2 , x a 12 a f(x) = (x + 1)2 , x b 13 a f(x) = x3 - 3x2 , x c
b f(x) = (x + 5)2 , x a b f(x) = (x - 3)2 , x b
b f(x) = x3 + 2x2 , x c
14 a f(x) = x3 - 3x, c x d 15 a f(x) = xex , x a
x
b f(x) = x3 - 12x, c x d b f(x) = xe -x, x a
x2 41 , b find an expression for f -1(x). 16 a f(x) = xe2, x a
b f(x) = xe− 3x, x a
x2
17 a f(x) = + 2 , x < b 4
18 b f(x) = 2 − 1 x < b 3− x
−
xx2 − (x ≠ :3)

For the function f(x) =

a find f -1(3)

19 Given that f(x) = 3e5x , find an expression for f -1(x). 1 y
5
20 The diagram shows the graph of y = f(x), where f(x) = x 3 − 3.

a Copy the graph and, on the same axes, sketch y = f -1(x).

b Find an expression for y = f -1(x).

x

x y

21 Let f(x) = e 12 − for all x ∈ . The diagram shows the graph of y = f(x). x
−1
a Sketch the graph of y = f -1(x).

b Find an expression for y = f -1(x) and state its domain.

322 14 Analysis and approaches: Functions

22 Given that f(x) = 2 + 1 for x > 2,
− 4
xx2

a Find an expression for f -1(x).

b State the range of f -1.

23 If f(x) = 1 , x > 0 and g(x) = 2x + 1, solve (g-1 f -1)(x) = 4.
4+ x

x y
x
24 Let f(x) = e2 + −x fo5r x ∈ . The diagram shows a part of the graph

of y = f(x).

a Copy the graph and sketch y = f -1(x) on the same axes.
b Find the exact solution of the equation f(x) = f -1(x).

−4

25 Let f(x) = x2 + 3 for x < a.
a Find the largest possible value of a such that f has an inverse function.
b For this domain, find an expression for f -1(x).

26 a For the function g(x) = 9(x - 5)2 , find the largest possible domain of the form x < k such that g has an inverse
function.

b For this domain, find an expression for g -1(x). +3
−4
27 A function f is called self-inverse if f(x) ≡ f -1(x). Find the value of a such that f(x) = axx is a self-inverse function.

Checklist

■ You should be able to form composite functions.

■ You should know when composite functions exist.

■ You should know what the identity function is and how it is related to inverse functions
(f f )(−)1 (fx = −1 f )(x) = x

■ You should be able to find inverse functions
To find an expression for f -1(x):

1 Let y = f(x)
2 Rearrange to make x the subject
3 State f -1(x) by replacing any ys with xs

■ You should know when inverse functions exist
A function has to be one-to-one to have an inverse.

Mixed Practice 323

■ Mixed Practice

1 a For the function f(x) 3=1−,xfind the inverse function, f -1(x).
b Verify that (f f )( −)1 x = x for all x.

2 A function is defined by h(x) = 5 − x for x a .
a State the largest possible value of a.
b Find h −(13) .

3 Given that f(x) e=3 x evaluate f (−41) .
4 The table shows some values of the function f( )x .

x 01234
f(x) 3 4 0 1 2

a Find (f f ) (2).
b Find f −(14).

5 Let f(x) = +x 3 and g(x) e=2 x . Solve the equation (g f )( )x1 = .
6 Given that g(x) 3=ln( x − 2),

a Find the largest possible domain for g.
b Using the domain for part a, find an expression for g ( −)1x and state its range.

7 Given the functions f(x) 3=1+x and g(x) x=, fi3nd (f g) ( −)1 x .

8 Let f(x) 2= x + 3 and g(x) = x3.
a Find (f g)( ) x .
b Solve the equation (f g)( ) x0.=

Mathematics SL November 2014 Paper 2 Q1

9 The following diagram shows the graph of =yf( ) x , for –4 .x 5
y

5

−5 5 x

−5
a Write down the value of

i f(−3);
ii f (1−)1.
b Find the domain of f -1.
c Sketch the graph of f -1.

Mathematics SL May 2014 Paper 1 TZ2 Q3