HOW TO TELL IF FUNCTION IS EVEN OR ODD FROM GRAPH

A Function can be classified as Even, Odd or Neither. This classification can be determined graphically or algebraically.

How to check if the graph is odd ?

The graph will be symmetric with respect to the origin.

In other words :

If you spin the picture upside down about the Origin, the graph looks the same!

How to check if the graph is even ?

The graph will be symmetric with respect to the y-axis.

Graphically determine whether the following functions are Even, Odd, or Neither

Problem 1 :

Solution :

The graph is symmetric about origin. So, it is odd function.

Problem 2 :

Solution :

The graph is not symmetric about origin, then it is not odd function.

The graph is not symmetric about y-axis, then it is not even function.

So, it is neither.

Problem 3 :

Solution :

Here y-axis is acting as a mirror. Clearly it is symmetric about y-axis. Then, it is even function.

Problem 4 :

Solution :

The graph is symmetric about origin. So, it is odd function.

Problem 5 :

Solution :

The graph is not symmetric about origin, then it is not odd function.

The graph is not symmetric about y-axis, then it is not even function.

So, it is neither.

Problem 6 :

Solution :

The graph is symmetric about origin. So, it is odd function.

Problem 7 :

Solution :

The graph is not symmetric about origin, then it is not odd function.

The graph is not symmetric about y-axis, then it is not even function.

So, it is neither.

Algebraically testing whether it is even odd or neither

For each of the following functions, classify each as: even, odd or neither. You must show your work to prove your classification.

Problem 8 :

f(x) = x² - 2x

Solution :

f(x) = x² - 2x

Put x = -x

f(-x) = (-x)² - 2(-x)

= x² + 2x

So, it is neither.

Problem 9 :

f(x) = 3x⁵ - 4x

Solution :

f(x) = 3x⁵ - 4x

Put x = -x

f(-x) = 3(-x)⁵ - 4(-x)

= -3x⁵ + 4x

Factoring the negative, we get

= -(3x⁵ - 4x)

f(-x) = -f(x)

So, it is odd function.

Problem 10 :

Solution :

So, it is even function.