Even and Odd Functions Examples

Is the function f(x) = x⁴ even?

Find f(x) and f(-x) and see if they're the same.

f(x) = x⁴
f(-x) = (-x)⁴ = x⁴

Yep, f(x) and f(-x) are the same, so this function is even.

Is the function f(x) = x³ + x odd?

Find f(x) and f(-x).

f(x) = x³ + x
f(-x) = (-x)³ + (-x) = -x³ – x

Since f(-x) is the negative of f(x), this function is odd.

Be Careful: When testing whether a function is even or odd, it's not good enough to check whether f(x) and f(-x) are the same at one specific number.
A coincidence might lead to the wrong answer.

Is the function f(x) = x³ – x even?

What happens when x = 1? We have

f(1) = (1)³ – (1) = 0
f(-1) = (-1)³ – (-1) = -1 + 1 = 0

The function f(x) is even, right? Nope. If we used x instead of the specific number 1 we would find

f(x) = x³ – x
f(-x) = -x³ + x.

These aren't the same except when x = ± 1 and when x = 0. The function f(x) isn't even after all.

Be Careful: When we're talking about functions, "even" and "odd" are not opposites. In contrast to integers, which must be either even or odd, a function might not be either one. There is only a loose connection between even and odd integers and even and odd functions.