Chapter 1.5, Problem 2QR

To determine

To evaluate: The value of expression $\left(g\circ f\right)\left(-7\right)$ .

Answer to Problem 2QR

The value of $\left(g\circ f\right)\left(-7\right)$ is $5$ .

Explanation of Solution

Given information: The functions are $f\left(x\right)=\sqrt[3]{x-1}$ and $g\left(x\right)=x^2+1$ .

Calculation:

Substitute $f\left(x\right)$ for $x$ in the function $g\left(x\right)=x^2+1$ .

  $\begin{array}{c}g\left(f\left(x\right)\right)={\left(f\left(x\right)\right)}^2+1\\ ={\left(\sqrt[3]{x-1}\right)}^2+1\end{array}$

Substitute $-7$ for $x$ in the above function.

  $\begin{array}{c}g\left(f\left(-7\right)\right)={\left(\sqrt[3]{-7-1}\right)}^2+1\\ ={\left(\sqrt[3]{-8}\right)}^2+1\\ ={\left(-2\right)}^2+1\\ =5\end{array}$

Therefore, the value of $\left(g\circ f\right)\left(-7\right)$ is $5$ .