Problem 9 - Precalculus Essentials

Question

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=-x \text { and } g(x)=-x$$

Step-by-Step Solution

Answer

The functions \(f(x) = -x\) and \(g(x) = -x\) are inverses of each other. \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 1: Find \(f(g(x))\)

We are given that \(f(x) = -x\) and \(g(x) = -x\). Substituting \(g(x)\) into \(f(x)\) gives: \(f(g(x)) = -(-x) = x\)

Step 2: Find \(g(f(x))\)

Similarly, substituting \(f(x)\) into \(g(x)\) gives: \(g(f(x)) = -(-x) = x\)

Step 3: Check if \(f\) and \(g\) are inverses

The functions \(f\) and \(g\) are inverses if both of the equalities \(f(g(x)) = x\) and \(g(f(x)) = x\) are true. After substituting, we find that both of these functions indeed equal \(x\), so \(f\) and \(g\) are inverses of each other.