# DETERMINE GRAPHICALLY OR ALGEBRAICALLY IF A FUNCTION IS ODD OR EVEN

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.

## Properties of odd and even functions :

Properties of odd function :

• The graph is symmetric about origin.
• The exponents of all terms in its equation are odd.

Properties of even function :

• The graph is symmetric about y-axis.
• The exponents of all terms in its equation are even.

## Algebraically Check if it is odd or even

A function f is even when, for each x in the domain of f,

f(-x) = f(x).

A function f is odd when, for each x in the domain of f,

f(-x) = -f(x).

## Graphically Checking If the Function is Odd or Even

Determine graphically using possible symmetry, whether the following functions are even, odd or neither.

Problem 1 :

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 2 :

Solution :

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

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 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.

## Problems on Algebraically Checking If It is Odd or Even

Verify algebraically whether each function is even, odd or neither!

Problem 1 :

f(x) = x3 - 6x

Solution:

f(x) = x3 - 6x

Put x = -x

f(-x) = (-x)3 - 6(-x)

= -x3 + 6x

So, it is neither.

Problem 2 :

g(x) = x4 - 2x2

Solution:

g(x) = x4 - 2x2

Put x = -x

g(-x) = (-x)4 - 2(-x)2

= x4 - 2x2

So, it is even function.

Problem 3 :

h(x) = x2 + 2x + 1

Solution:

h(x) = x2 + 2x + 1

Put x = -x

h(-x) = (-x)2 + 2(-x) + 1

= x2 - 2x + 1

h(-x) ≠ h(x)

So, it is neither.

Problem 4 :

f(x) = x2 + 6

Solution:

f(x) = x2 + 6

Put x = -x

f(-x) = (-x)2 + 6

= x2 + 6

f(-x) = f(x)

So, it is even function.

Problem 5 :

g(x) = 7x3 - x

Solution:

g(x) = 7x3 - x

Put x = -x

g(-x) = 7(-x)3 - (-x)

= -7x3 + x

g(-x) ≠ g(x)

So, it is odd function.

Problem 6 :

h(x) = x5 + 1

Solution:

h(x) = x5 + 1

Put x = -x

h(-x) = (-x)5 + 1

= -x5 + 1

h(-x) ≠ h(x)

So, it is neither.

Problem 7 :

f(x) = x√(4 - x2)

Solution:

f(x) = x√(4 - x2)

Put x = -x

f(-x) = (-x)√(4 - (-x)2)

= -x√(4 - x2)

So, it is neither.

Problem 8 :

g(x) = x4√(1 + x)

Solution:

g(x) = x4√(1 + x)

Put x = -x

g(-x) = (-x)4√(1 + (-x))

= x4√(1 - x)

So, it is neither.

Problem 9 :

h(x) = |x| - 1

Solution:

h(x) = |x| - 1

Put x = -x

h(-x) = |-x| - 1

= x - 1

So, it is even function.

Problem 10 :

Solution:

So, it is even function.

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