Question

How do you tell whether a function is even, odd or neither?

Answer 1

To determine this, plug `-x` in for `x` and see what happens.

Explanation:

The first step is to replace `x` with `–x`. In other words, calculate `f(-x)`.

If the function doesn't change (i.e. `f(-x) = f(x)`. then it is even. For instance, `f(x) = x^2` is even because #f(-x) = (-x)^2 = x^2.

If the function is the reverse of what it was originally (i.e. `f(-x) = -f(x)`, then it is odd. For instance, `f(x) = x` is odd because `f(-x) = -x = -f(x)`.

If anything else happens, the function is neither even nor odd. For instance, `f(x) = x^2 + x` is neither even nor odd because `f(-x) = (-x)^2 + -x = x^2 - x`, and that is neither the function we started with, nor the reverse.