Chapter 1, Problem 30SGA

To determine

To find: $\left[f\circ g\right]\left[x\right]$.

Answer to Problem 30SGA

  $\left\{x|x\le -11\right\}$

Explanation of Solution

Given information: The functions given are as follows:

  $\begin{array}{l}f\left(x\right)=\sqrt{x-16}\\ g\left(x\right)=5-x\end{array}$

Calculation:

Put the values of $g\left(x\right)$ in $f\left(g\left(x\right)\right)$ in the equation below and simplify as follows:

  $\begin{array}{c}\left[f\circ g\right]\left[x\right]=f\left(g\left(x\right)\right)\\ =\sqrt{\left(g\left(x\right)\right)-16}\\ =\sqrt{\left(5-x\right)-16}\left[\text{Substitute }g\left(x\right)\right]\\ =\sqrt{-x-11}\end{array}$

Thus, the value of $\left[f\circ g\right]\left[x\right]$ is $\sqrt{-x-11}$ .

The square root cannot be taken negative or end up with real number. So, put the square root greater than or equal to zero and evaluate x as shown below.

  $\begin{array}{c}\sqrt{-x-11}\ge 0\\ -x-11\ge 0\left[\text{Taking square root both the sides}\right]\\ -x\ge 11\\ x\le -11\end{array}$

Therefore, the domain is $\left\{x|x\le -11\right\}$ .