How to tell if a function is even, odd or neither?

A function y = f(x) is an

even function of x if f(-x) = f(x)

odd function of x if f(-x) = -f(x)

for every x in the function’s domain.

The names even and odd comes from the powers of \boldsymbol{x}.

Even function:

If y is an even power of x as in y = x^{2} and y = x^{4} , it is an even function of x because

(-x)^{2} = x^{2}

(-x)^{4} = x^{4}

Odd function:

If y is an odd power of x as in y = x and y = x^{3} , it is an odd function of x because

(-x)^{1} = -x

(-x)^{3} = -x^{3}

Characteristics of an even function:

1. The graph of an even function is symmetric about the \boldsymbol{y}-axis.

This is because in the case of an even function the value of y is same for x and -x.

Graph to show even function's symmetry about y-axis

2. The graph of an even function does not change if we reflect it across the \boldsymbol{y}-axis.

Graph demonstrating even function reflected across y-axis

Characteristics of an odd function:

1. The graph of an odd function is symmetric about the origin.

This is because in the case of an odd function the value of y at -x is the negative of the value of y at x.

Graph to show odd function's symmetry about the origin

2. The graph of an odd function does not change if we rotate it by \boldsymbol{180º} about the origin.

 

3. An odd function always takes the value \boldsymbol{0} at \boldsymbol{x = 0}.

This is because for an odd function, f(-x) = -f(x)

At x = 0, f(-0) = -f(0)

 \implies f(0) = -f(0)

\implies 2f(0) = 0 \implies f(0) = 0

Therefore, if a function is odd, f(0) always equals 0, but f(0) = 0 does not imply that the function is odd.

4. The graph of an odd function always passes through the origin.

This is because for an odd function, f(0) = 0 .

Let’s see some examples:

Example 1: Consider a function, \boldsymbol{f(x) = x^{2} + 1}. Let’s see whether this function is even, odd or neither.

Solution:

To check whether the given function is odd or even, first, we need to determine f(-x) . For that, we will substitute -x in place of x in f(x).

f(-x) = (-x)^{2}+1

= x^{2}+1 = f(x)

Since \boldsymbol{f(-x) = f(x)}, the given function is even.

Below is the graph of this function and we can see that it is symmetric about the y-axis.

Even and Odd Functions Example 1 graph

Example 2: Consider a function, \boldsymbol{f(x) = x + 1}. Determine whether this function is even, odd or neither.

Solution:

As discussed earlier, first, we need to determine f(-x) .

f(-x) = (-x) + 1 = -x + 1

-f(x) = – (x + 1) = -x – 1

Clearly, f(-x) \neq f(x) and f(-x) \neq -f(x)

Therefore, the given function is neither an even function nor an odd function.

Below is the graph of this function and we can see that it is neither symmetric about the y-axis nor it is symmetric about the origin.

Even and Odd Functions Example 2 graph

Example 3: Consider a function, \boldsymbol{f(x) = x^{3} + x}. Determine whether this function is even, odd or neither.

Solution:

First, we need to determine f(-x) .

f(-x) = (-x)^{3} + (-x)

= -x^{3} – x

= -(x^{3} + x) = -f(x)

Since, \boldsymbol{f(-x) = -f(x)}, the given function is odd.

Below is the graph of this function and we can see that it is symmetric about the origin.

Even and Odd Functions Example 3 graph

Example 4: Consider a function, \boldsymbol{f(x) = 0}. Determine whether this function is even, odd or neither.

Solution:

First, we need to determine f(-x) .

Since f(x) is a constant function, therefore, f(-x) also equals 0

f(-x) = 0

Also, -f(x) = 0

Clearly, f(-x) = f(x) and f(-x) = -f(x)

Therefore, the given function is both an even function and an odd function. 

The graph of this function lies along the x-axis and it is symmetric about the origin as well as the y-axis.

Even and Odd Functions Example 4 graph

\boldsymbol{f(x) = 0} is the only real function which is both an even function and an odd function.

Some Other Properties:

1. The sum of two or more even functions is also an even function.

2. The sum of two or more odd functions is also an odd function.

3. The sum of an even function and an odd function is neither an even function nor an odd function unless one of the functions is zero.

4. The product of two or more even functions is also an even function.

5. The product of two odd functions is an even function.

6. The product of three odd functions is an odd function.

7. The product of an even function and an odd function is an odd function.

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Even and Odd Functions
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