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Chapter 0: Problem 9

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

Short Answer

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

Step by step solution

01

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 \(h(x) = g(f(x))\).
02

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: \[ h(x) = g(f(x)) = 3(\sqrt[3]{x^{2}-1})^3 + 1. \]
03

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 \(h(2) = 10\).

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Most popular questions from this chapter

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=x^{2}-0.1 x $$

You are given the graph of a function \(f .\) Determine whether \(f\) is one-to- one.

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Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=x^{2}, \quad y=\left|x^{2}-1\right|\)
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