Chapter 6.3, Problem 25PPS

To determine

The domain and range of the function $f\left(x\right)=2\sqrt{x-5}-6$ . And sketch the graph of the function.

Answer to Problem 25PPS

The domain of the function is $\left\{x|x\ge 5\right\}$ and range of the function is $\left\{y|y\ge -6\right\}$ .

Explanation of Solution

Given information: The given function is,

  $f\left(x\right)=2\sqrt{x-5}-6\mathrm{....}\left(1\right)$

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

  $\begin{array}{c}x-5\ge 0\\ x\ge 5\end{array}$

Hence the domain of the function is $\left\{x|x\ge 5\right\}$ .

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),

  $\begin{array}{c}f\left(5\right)=2\sqrt{\left(5\right)-5}-\mathrm{6......}\left(\text{For }x=5\right)\\ =-6\end{array}$

Hence the range of the function is $\left\{y|y\ge -6\right\}$

Graph: The graph for the function is shown as,