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

1 Answer
Sep 13, 2015

Answer:

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.