Question

Show that \(f\) and \(g\) are inverse functions (a analytically and (b) graphically. $$ f(x)=x^{3}, \quad g(x)=\sqrt[3]{x} $$

Short answer

It is shown that the functions \(f(x) = x^{3}\) and \(g(x) = \sqrt[3]{x}\) are inverses of each other as the compositions \(f(g(x))\) and \(g(f(x))\) both equal \(x\), and the graph of \(f\) is the reflection of the graph of \(g\) over the line \(y = x\).

Step-by-step solution

Test the composition for \(f(g(x))\)

First, let's substitute \(g(x) = \sqrt[3]{x}\) into function \(f(x) = x^{3}\). \n \n\nSubstituting gives \(f(g(x)) = (\sqrt[3]{x})^{3}\). Simplifying this expression gives us \(x\). This means that \(f(g(x)) = x\) is true for all \(x\).

Test the composition for \(g(f(x))\)

Secondly, let's substitute \(f(x) = x^{3}\) into function \(g(x) = \sqrt[3]{x}\). \n\nSubstituting into \(g(x)\) gives \(g(f(x)) = \sqrt[3]{(x^{3})}\). Simplifying this expression gives us \(x\). This means \(g(f(x)) = x\) is true for all \(x\).

Graphically show that \(f\) and \(g\) are inverse functions

Graphically showing that two functions are inverses of each other involves reflecting one function over the line \(y = x\) and checking if it coincides with the other function. The cubic function \(f(x) = x^{3}\) reflects perfectly over the line \(y = x\) to give the cubic root function \(g(x) = \sqrt[3]{x}\). Thus graphically we can also conclude that these functions are inverse of each other.