Chapter 6.3, Problem 28PPS

To determine

The domain and range of the function $f\left(x\right)=-3\sqrt{x+7}+9$ . And sketch the graph of the function.

Answer to Problem 28PPS

The domain of the function is $\left\{x|x\ge -7\right\}$ and range of the function is $\left\{y|y\le 9\right\}$ .

Explanation of Solution

Given information: The given function is,

  $f\left(x\right)=-3\sqrt{x+7}+9\mathrm{....}\left(1\right)$

Calculation: The domain of the function is defined for which,

  $\begin{array}{c}x+7\ge 0\\ x\ge -7\end{array}$

Hence the domain of the function is $\left\{x|x\ge -7\right\}$ .

And range of the function can be evaluated by substituting the lower limit of domain in the function for which the function is defined. Therefore from (1),

  $\begin{array}{c}f\left(-7\right)=-3\sqrt{\left(-7\right)+7}+\mathrm{9......}\left(\text{For }x=-7\right)\\ =9\end{array}$

Also the function $f\left(x\right)=-3\sqrt{x+7}+9$ is a decreasing function. Therefore $f\left(-7\right)=9$ is the upper limit of range.

Hence the range of the function is $\left\{y|y\le 9\right\}$ .

Graph: The graph for the function is shown as,