Topics
2. Functions and Relations
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AustraliaNSW
Year 11

2.04 Odd and even functions

Lesson
Worksheet
Practice
Worksheet
Identifying odd and even functions
1

For each graph below, identify which type of function it is: odd, even or neither.

a
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
b
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
c
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
d
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
e
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
2

Consider the graph below:

a

Find the value of y when:

i
x = 4
ii
x = - 4
b

How can the part of the graph for x < 0 be obtained from the part for x > 0?

c

Determine whether the function is odd, even or neither.

-6
-4
-2
2
4
6
x
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
y
3

Consider the graph below:

a

Find the value of y when:

i
x = 1
ii
x = - 1
b

How can the part of the graph for x < 0 be obtained by transforming the part of the graph for x > 0?

c

Is the function odd, even or neither?

-3
-2
-1
1
2
3
x
-3
-2
-1
1
2
3
y
4

Consider the graph below:

a

Find the value of y when:

i
x = - 1
ii
x = 1
b

Is the function is odd, even or neither?

-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
1
2
3
4
5
y
5

For each function below:

i

Complete a table of values:

x- 3- 2-10123
y
ii

Determine whether the function is odd or even.

a
y = x^{3} - 4 x
b
y = 2^{x} + 2^{ - x }
6

For each of the following tables, determine if f \left( x \right) is even, odd or neither.

a
xf(x)
-6 10
-47
-23
02
2-3
4-7
6-10
b
xf(x)
-66
-45
-23
00
2-3
4-5
6-6
7

For each function below:

i

Find f \left( -x \right).

ii

State whether f \left( x \right) is even, odd or neither?

a
f \left( x \right) = - 2 x^{4} - 6 x^{2}
b
f \left( x \right) = 9 x^{3} - 3 x
c
f \left( x \right) = - 5 x^{7} - 9 x^{3} + 3 x
d
f \left( x \right) = \dfrac{1}{6 x}
e
f \left( x \right) = x^{2} + x
f
f \left( x \right) = 7
g
f \left( x \right) = 9 x^{2} - 6 x
h
f \left( x \right) = x^{3} - 5 x
i
f \left( x \right) = \dfrac{1}{x^{2} - 9}
j
f \left( x \right) = x^{4} + 8 x^{3} - 7 x^{2}
k
f \left( x \right) = 8 x - \dfrac{5}{x}
l
f \left( x \right) = 4^{x}
m
f \left( x \right) = \dfrac{x - 5}{x + 5}
n
f \left( x \right) = \sqrt{2 - x}
o
f \left( x \right) = \sqrt{x^{8} - 8}
p
f \left( x \right) = \dfrac{4}{\sqrt{36 - x^{2}}}
q
f \left( x \right) = \dfrac{1}{x^{3} - 1}
r
f \left( x \right) = \dfrac{3 x}{\sqrt{81 - x^{2}}}
Examples of odd and even functions
8

Complete each table given that:

a

f \left( x \right) is an even function:

x-6-4-2246
f(x)-18-2227
b

f \left( x \right) is an odd function:

x-9-6-30369
f(x)-6-3-7
9

The following function has been partially graphed below for x\leq0. If this function is an odd function, sketch the complete graph of the function.

-4
-3
-2
-1
1
2
3
4
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
10

The following function has been partially graphed below for x\leq0. If this function is an even function sketch the complete graph of the function.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
11

For what values of n, where n is a positive integer, is the power function f \left( x \right) = k x^{n}:

a

An even function?

b

An odd function?

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MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

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