Problem 70 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

Two functions \(f\) and \(g\) are inverses of each other if the composition of \(f\) and \(g\) in both orders (i.e., \(f(g(x))\) and \(g(f(x))\)) results in the original input variable \(x\).

Step 1: Understanding Function Composition

Given two functions \(f(x)\) and \(g(x)\), their composition is denoted as \(f(g(x))\) or \(g(f(x))\). This means that the output of one function becomes the input of the other.

Step 2: Checking for Inverse - First Condition

To confirm whether \(f(x)\) and \(g(x)\) are inverses, apply the first condition - compute \(f(g(x))\). If \(f(g(x)) = x\), we can say that the first condition is met.

Step 3: Checking for Inverse - Second Condition

The second condition to validate inverses is to compute \(g(f(x))\). If \(g(f(x)) = x\), we can say that the second condition is met. Both conditions must be satisfied for two functions to be inverses of each other.

Step 4: Result Evaluation

If and only if both conditions are satisfied (i.e. \(f(g(x)) = x\) and \(g(f(x)) = x\)), then we indicate that function \(f\) and function \(g\) are inverses of each other. Consequently, this means that function \(f\) undoes the operation of function \(g\), and vice versa.