Chapter 7.3, Problem 2ACYP

To determine

To find: The domain and the range of given function and plot the graph.

Answer to Problem 2ACYP

The domain is $\left\{x:x\ge -6\right\}$ and the range is $\left\{y:y\ge 0\right\}$ .

Explanation of Solution

Given:

The given function is:

  $f\left(x\right)=2\sqrt{x+4}\dots \left(1\right)$

The value of what’s under the square root must be greater than equal to zero.

Therefore,

  $\begin{array}{c}x+4\ge 0\\ x\ge -4\end{array}$

So the domain is $\left\{x:x\ge -4\right\}$ .

When,

  $x=-4$ ,

Further simplified as:

  $\begin{array}{c}f\left(-4\right)=2\sqrt{-4+4}\\ =0\end{array}$ .

And

  $\begin{array}{c}\underset{x\to +\infty }{\lim }f\left(x\right)=\underset{x\to +\infty }{\lim }2\sqrt{x+4}\\ =+\infty \end{array}$

Therefore the range is $\left\{y:y\ge 0\right\}$ .

The table for the graph of the function is given as:

$x$	$-4$	$0$	$12$
$f\left(x\right)$	$0$	$4$	$8$

The graph of the given function is given by: