Question

Consider the functions \(f, g: \mathbb{R} \rightarrow \mathbb{R}\) defined as \(f(x)=\sqrt[3]{x+1}\) and \(g(x)=x^{3}\). Find the formulas for \(g \circ f\) and \(f \circ g\).

Short answer

The formulas for the function compositions are \(g \circ f(x) = x + 1\) and \(f \circ g(x) = \sqrt[3]{x^{3}+1}\).

Step-by-step solution

Define the functions

The functions \(f\) and \(g\) are given as \(f(x)=\sqrt[3]{x+1}\) and \(g(x)=x^{3}\) respectively.

Find the formula for \(g \circ f\)

The composition of the functions \(g \circ f\) means that \(f\) is substituted into \(g\). Therefore, it's calculated as \(g(f(x))\). Substituting the expression for \(f(x)\) into \(g(x)\), we have \(g(f(x)) = (\sqrt[3]{x+1})^{3}\), which simplifies to \(g(f(x)) = x + 1\). Therefore, the formula for \(g \circ f\) is \(x + 1\).

Find the formula for \(f \circ g\)

The composition of the functions \(f \circ g\) means that \(g\) is substituted into \(f\). Therefore, it's calculated as \(f(g(x))\). Substituting the expression for \(g(x)\) into \(f(x)\), we have \(f(g(x)) = \sqrt[3]{(x^{3}+1)}\). Therefore, the formula for \(f \circ g\) is \(\sqrt[3]{x^{3}+1}\).