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