Chapter 7, Problem 34SGR

To determine

To Graph and Find :

Graph the given function and find the domain and range.

Answer to Problem 34SGR

  $\begin{array}{l}\text{The domain of the function is {x | x}\ge -5\text{} and }\\ \text{the range of the function is {f(x) | f(x) }\ge -3\text{ }}\text{.}\end{array}$

Explanation of Solution

Given:

The function is:

  $f(x)=\sqrt{x+5}-3$

Concept Used:

• Definition of nth root:

• Words : For any real numbers a and b , and any positive integer n, if $a^n=b$ then a is the nth root of b. That is $a=\sqrt[n]{b}$
• $\begin{array}{l}If\text{ n is even: }\\ \sqrt[n]{b}\ge 0\text{ and }b\ge 0\end{array}$
• $\begin{array}{l}\sqrt[n]{b}\\ Here\\ b=radicand\\ n=index\\ \sqrt{}=radicalsign\end{array}$

Calculation:

  $\begin{array}{l}Given\\ f(x)=\sqrt{x+5}-3\\ \text{Since the index is 2, which is even; so the radicand is non-negative}\text{.}\\ x+5\ge 0\\ x\ge -\mathrm{5..............}[subtract\text{ 5 from each side]}\\ \text{Make a table of values for x}\ge -5\text{ and corresponding y-values}\text{.}\end{array}$

x	$y=\sqrt{x+5}-3$
-5	-3
-4	-2
-3	-1.59
-2	-1.27
-1	-1
0	-0.76

Plot the points and join to form the graph of the given function:

  

  $\begin{array}{l}So,\text{ the domain of the function is {x | x}\ge -5\text{} and }\\ \text{the range of the function is {f(x) | f(x) }\ge -3\text{ }}\text{.}\end{array}$