Problem 47 - College Algebra

Question

Explain how to determine if two functions are inverses of each other.

Step-by-Step Solution

Answer

Two functions \(f(x)\) and \(g(x)\) are inverses of each other if the compositions \(f(g(x)) = x\) and \(g(f(x)) = x\). Compose the functions in both orders, simplify the results, and check whether these results equal \(x\).

Step 1: Understanding the concept of inverse functions

In mathematics, function \(f\) and function \(g\) are inverses of each other if the following is true for every \(x\) in their domains: \(f(g(x)) = x\) and \(g(f(x)) = x\). The first function undoes the effect of the second, and vice versa. This is the condition we need to verify to determine if two given functions are inverses of each other.

Step 2: Composing the functions

Given two functions, \(f(x)\) and \(g(x)\), compose the functions in both orders. First, compute \(f(g(x))\) and then \(g(f(x))\). in other words, plug \(g(x)\) into the function \(f(x)\) and then plug \(f(x)\) into the function \(g(x)\).

Step 3: Simplifying the compositions

After performing each composition, simplify the results. With most functions, this will involve performing operations like combining like terms, distributing, or factoring.

Step 4: Comparing the compositions to the identity function

Check if the simplified compositions equal \(x\). If the compositions \(f(g(x))\) and \(g(f(x))\) both equal \(x\), then \(f(x)\) and \(g(x)\) are inverses of each other. If either composition does not simplify to \(x\), then the two functions are not inverses of each other.