# Odd and Even Functions

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```					                             Symmetric about the y axis

FUNCTIONS
Symmetric about the origin
Even functions have y-axis Symmetry
8
7
6
5
4
3
2
1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7

So for an even function, for every point (x, y) on
the graph, the point (-x, y) is also on the graph.
Odd functions have origin Symmetry
8
7
6
5
4
3
2
1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7

So for an odd function, for every point (x, y) on the
graph, the point (-x, -y) is also on the graph.
x-axis Symmetry
We wouldn’t talk about a function with x-axis symmetry
because it wouldn’t BE a function.

8
7
6
5
4
3
2
1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
A function is even if f( -x) = f(x) for every number x in
the domain.

So if you plug a –x into the function and you get the
original function back again it is even.

f x   5 x  2 x  1
4           2         Is this function even?
YES
f  x   5( x)  2( x)  1  5 x  2 x  1
4            2            4          2

f x   2 x  x 3
Is this function even?
NO
f  x   2( x)  ( x)  2 x  x
3                         3
A function is odd if f( -x) = - f(x) for every number x in
the domain.

So if you plug a –x into the function and you get the
negative of the function back again (all terms change signs)
it is odd.

f x   5 x  2 x  1
4           2         Is this function odd?
NO
f  x   5( x)  2( x)  1  5 x  2 x  1
4            2            4         2

f x   2 x  x 3
Is this function odd?
YES
f  x   2( x)  ( x)  2 x  x
3                         3
If a function is not even or odd we just say neither
(meaning neither even nor odd)

Determine if the following functions are even, odd or
neither.
Not the original and all
f x   5 x  1
3
terms didn’t change
signs, so NEITHER.

f  x   5 x   1  5 x  1
3           3

f x   3x  x  24       2           Got f(x) back so
EVEN.

f  x   3( x)  ( x)  2  3x  x  2
4       2               4       2
Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.

Stephen Corcoran