Home › Functions and graphs

How to tell whether a function is even, odd or neither

The rule: substitute -x into the whole function

A function is even when $f (- x) = f (x)$ for every $x$ in its domain, and odd when $f (- x) = - f (x)$ . If neither identity holds, the function is neither. Keep parentheses around $- x$ and negate the entire expression when you calculate $- f (x)$ .

Problem

Determine algebraically whether each function is even, odd or neither. Example 1. $g (x) = 9 x^{3} + 5$ . Example 2. $f (x) = 9 2 x$ .

Answer

g (x) = 9 x^{3} + 5

is neither even nor odd.

f (x) = 9 2 x

is odd.

Both identities were compared symbolically, not inferred only from the appearance of the formulas.

Step-by-step solution

Example 1: neither

Substitute $- x$ for $x$ :

g (- x) = 9 (- x)^{3} + 5 = - 9 x^{3} + 5

This is not equal to $g (x) = 9 x^{3} + 5$ , so the function is not even. Now negate the entire original function:

- g (x) = - (9 x^{3} + 5) = - 9 x^{3} - 5

Because $g (- x) = - 9 x^{3} + 5$ is also not equal to $- g (x)$ , the function is neither even nor odd. The constant vertical shift $+ 5$ breaks the odd symmetry of $9 x^{3}$ .

Example 2: odd

An odd root preserves the sign of its input:

f (- x) = 9 - 2 x = - 9 2 x = - f (x)

The domain is all real numbers and is symmetric about zero, so the identity holds throughout the domain. Therefore $f$ is odd.

How to check yourself

Do not classify a function by individual terms. A mixture such as an odd term plus a nonzero constant is usually neither. Also distinguish $f (- x)$ from $- f (x)$ : one changes the input, while the other negates the complete output.

Add to Chrome - solve your own

Screenshot any problem on your screen and get the steps. Chrome on a computer; free.