Question

Evaluate \(h(2)\), where \(h=g \circ f\). \(f(x)=\sqrt[3]{x^{2}-1}, \quad g(x)=3 x^{3}+1\)

Short answer

The short answer is: \(h(2)=10\).

Step-by-step solution

Write the expression for the composite function \(h(x) = g(f(x))\)

A composite function is a function obtained by composing two functions. In our case, we have \(h(x) = g \circ f\), which means that the output of \(f(x)\) is used as an input for \(g(x)\). This can be written as \(h(x) = g(f(x))\).

Substitute the given functions \(f(x)\) and \(g(x)\) into the expression for \(h(x)\)

We are given the functions \(f(x)=\sqrt[3]{x^{2}-1}\) and \(g(x)=3x^3+1\). We can substitute these into our expression for \(h(x)\) as follows: \[ h(x) = g(f(x)) = 3(\sqrt[3]{x^{2}-1})^3 + 1. \]

Evaluate \(h(2)\) by plugging \(2\) for the input value \(x\)

To find the value of \(h(2)\), we replace \(x\) with \(2\) and perform the calculations: \begin{align*} h(2) &= 3(\sqrt[3]{(2)^{2}-1})^3 + 1 \\ &= 3(\sqrt[3]{3})^3 + 1 \\ &= 3(3) + 1 \\ &= 9 + 1 \\ &= 10. \end{align*} So, we find that \(h(2) = 10\).