Problem 70 - College Algebra

Question

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

Step-by-Step Solution

Answer

Two functions are inverses of each other if the composition of one function with the other in both directions results in the identity output, i.e. the original input.

Step 1: Understand the Concept of Inverse Functions

An inverse function is a function that ‘reverses’ the work of the original function. In other words, if function f applied to an input \(x\) gives a result \(y\), then applying its inverse function \(g\) to \(y\) gives the result \(x\) again. Formally, if the pair \((x, y)\) is on the graph of \(f\), then the pair \((y, x)\) is on the graph of \(g\).

Step 2: Constructing the Composite Functions

The primary step in proving two functions as inverses of each other is to construct the composite function. A composite function is created when one function is substituted into another. So, for given functions \(f(x)\) and \(g(x)\), form \(f(g(x))\) and \(g(f(x))\).

Step 3: Simplify the Composite Functions

Next, simplify both \(f(g(x))\) and \(g(f(x))\). This means replacing \(g(x)\) in function \(f\) and \(f(x)\) in function \(g\) and simplifying the resulting expressions.

Step 4: Equality Check

Once the simplification is done, check if \(f(g(x)) = x\) and \(g(f(x)) = x\). If both equalities hold true, then the two functions \(f\) and \(g\) are indeed inverses of each other.