Chapter 6.3, Problem 2BGP

To determine

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

Answer to Problem 2BGP

The domain is $\left\{x:x\ge 5\right\}$ and the range is $\left\{y:y\ge 3\right\}$ .

Explanation of Solution

Given:

The given function is:

  $f\left(x\right)=\frac{1}{4}\sqrt{x-5}+3\dots \left(1\right)$

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

Therefore,

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

So the domain is $\left\{x:x\ge 5\right\}$ .

When,

  $x=5$ ,

Further simplified as:

  $\begin{array}{c}f\left(5\right)=\frac{1}{4}\sqrt{5-5}+3\\ =3\end{array}$ .

And

  $\begin{array}{c}\underset{x\to +\infty }{\lim }f\left(x\right)=\underset{x\to +\infty }{\lim }\frac{1}{4}\sqrt{x-5}+3\\ =+\infty \end{array}$

Therefore the range is $\left\{y:y\ge 3\right\}$ .

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

$x$	$5$	$21$
$f\left(x\right)$	$3$	$4$

The graph of the given function is given by: