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

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

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.