Problem 122 - Precalculus Essentials

Question

If you are given a function's equation, how do you determine if the function is even, odd, or neither?

Step-by-Step Solution

Answer

To check if a function \(f(x)\) is even, odd or neither, replace \(x\) by \(-x\) in the function and simplify. If \(f(-x) = f(x)\) then the function is even, if \(f(-x) = -f(x)\) the function is odd. If neither conditions is true then the function is neither even nor odd.

Step 1: Replace x with -x

Given a function \(f(x)\), compute \(f(-x)\) by substituting \(-x\) for every occurrence of \(x\).

Step 2: Compare to Original

If \(f(-x) = f(x)\) for all \(x\) in the domain, the function is even.
If \(f(-x) = -f(x)\) for all \(x\) in the domain, the function is odd.
If neither condition holds, the function is neither even nor odd.