SM015 - 5.1 (Notes Solutions)

5.1 Functions

Friday, July 10, 2020 3:14 PM

Learning Outcomes:
At the end of the lesson, students should be able to
(a) Define a function.
(b) Identify a function from the graph by using vertical line test
(c) Identify a one-to-one function by using algebraic approach

or horizontal line test.
(d) Sketch the graph of a function.
(e) State the domain and range of a function.

5.1 Functions Page 1

Functions

Monday, June 29, 2020 10:57 AM

Functions
 A function is defined as a relation in which every elements in

the domain has a unique image element in the codomain.

In other words, a function is either one-to-one relation or many-
to-one relation.

One-to-One Many-to-One

 A function can be identify by plotting graph based on all the
ordered pairs of the relation and then apply Vertical Line Test.

Onto Functions
 A function which each element of the codomain is mapped to

at least one element of the domain.

Remarks:
• "Onto" is an important criterion to determine the existence of inverse for a function.
• Most of the functions in the syllabus are considered as onto function.

One-to-One Functions
 A function which each element of the codomain is mapped to

exactly one element of the domain.

5.1 Functions Page 2

Vertical Line Test

Monday, June 29, 2020 1:30 PM

 The Vertical Line Test is used to determine a relation is whether
a function or not based on graph.

 The test is carried out by observing the all vertical lines parallel
to axis that intersects the graph.
• The graph is a function if all the vertical lines cuts the graph
at ONLY one point
E.g.

By Vertical Line Test, is a function because all the
vertical lines cuts the graph of at ONLY one point.
• The graph is NOT a function if there exist a vertical line cuts
the graph at MORE THAN one point.
E.g.

By Vertical Line Test, is NOT a function because there
exists a vertical line cuts the graph of at MORE THAN
one point.

5.1 Functions Page 3

One-to-One function

Monday, June 29, 2020 1:30 PM

 A one-to-one function is a function which each element of the
codomain is mapped to exactly one element of the domain.

 A one-to-one function can be identify by 2 methods, i.e.

(a) Algebraic method

• If implies for ,

then is one-to-one.

E.g.
Let

is one-to-one because implies for

• If DOES NOT implies [i.e. ]
for
, then is NOT one-to-one.
E.g.
Let

or

is NOT one-to-one because DOES NOT

implies for

(b) Graphical method
• Apply Horizontal Line Test on the graph of the function.

5.1 Functions Page 4

Horizontal Line Test

Monday, June 29, 2020 1:30 PM

 The Horizontal Line Test is used to determine a function is
whether one-to-one or not based on graph.

 The test is carried out by observing the all horizontal lines
parallel to axis that intersects the graph.
• The function is One-to-One if all the horizontal lines cuts
the graph at ONLY one point
E.g.

By Horizontal Line Test, is one-to-one because all the
horizontal lines cuts the graph of at ONLY one point.
• The function is NOT One-to-One if there exist a horizontal
line cuts the graph at MORE THAN one point.
E.g.

By Horizontal Line Test, is NOT one-to-one because
there exists a horizontal line cuts the graph of at
MORE THAN one point.

5.1 Functions Page 5

Example 1

Tuesday, June 30, 2020 4:01 PM

Determine whether the following is one-to-one function or not.

,

Algebraic Method: Graphical Method:

Let https://www.desmos.com/calculator/fgczwid0v0

Since does not
imply for

is NOT a one-to-one function.
By Horizontal Line Test, is NOT a
one-to-one function

[There exists a horizontal line cuts the graph
of at MORE THAN one point.]

Algebraic Method: Graphical Method:
Let
https://www.desmos.com/calculator/dyfvtzo1y7

Since implies
for

is a one-to-one function

By Horizontal Line Test, is a
one-to-one function

[All the horizontal lines cut the graph of
at ONLY one point.]

5.1 Functions Page 6

Domain & Range of a Function

Monday, June 29, 2020 10:57 AM

 If is a function that maps elements of set to elements of set ,
then
• Set is the domain of . [denoted by ]

• Set is the codomain of .
• The set of images in is the range of . [denoted by ]

E.g.

Domain Codomain Domain
Codomain Codomain
Range Range

Note: Range Note: Range = Codomain

Remarks:

• Range is the subset of (or equal to) Codomain.
• If , then is a onto function.

Domain of a function , Range of a function ,

• the set of all real values of • the output of based on the
such that is defined. domain of .

• Refer to the interval of . • Refer to the interval of .

Both domain & range of a function can be obtained by 2 methods, i.e.
(a) Graphical method

• Domain the coverage of the graph of on

• Range the coverage of the graph of on

(b) Algebraic method to be
• By calculation based on the conditions for the function
defined ,
Hints:





5.1 Functions Page 7

Example 1 (a) - (d)

Tuesday, June 30, 2020 4:01 PM

Find the domain and range of the following function by using graph.

Domain = Domain =
Range = Range =

Domain = Domain =
Range = Range =

5.1 Functions Page 8

Example 1 (e) - (g)

Tuesday, June 30, 2020 4:01 PM

Find the domain and range of the following function by using graph.

Domain = Domain =
Range = Range =

Domain = Domain =
Range = Range =

5.1 Functions Page 9

Example 2

Tuesday, June 30, 2020 4:01 PM

Sketch the graph of the functions and find the domain and range for
each of the function given.

https://www.desmos.com/calculator/e2ojpc3bqc

Algebraic method:

Domain = Domain:
Range = is defined for all .

Domain = Range:
Range =
https://www.desmos.com/calculator/7rns4pfqcl

Algebraic method:
Domain:

is defined for all .
Range:

5.1 Functions Page 10

Example 3

Tuesday, June 30, 2020 4:01 PM

Sketch the graph of the functions and find the domain and range for
each of the function given.

https://www.desmos.com/calculator/8khkgsmklv

Algebraic method:
Domain:

Range:

Domain =
Range =

https://www.desmos.com/calculator/ilsmfprcfo

Algebraic method:
Domain:

Range:

Domain =
Range =

5.1 Functions Page 11