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QUESTION

# 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.