Chapter 7, Problem 36SGR

To determine

To Graph and Find :

Graph the given function and find the domain and range.

Answer to Problem 36SGR

  $\begin{array}{l}\text{The domain of the function is {x | x}\ge -4\text{} and }\\ \text{the range of the function is {f(x) | f(x) }\le \text{-1 }}\text{.}\end{array}$

Explanation of Solution

Given:

The function is:

  $f(x)=-\frac{1}{3}\sqrt{x+4}-1$

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:

  $\begin{array}{l}Given\\ f(x)=-\frac{1}{3}\sqrt{x+4}-1\\ \text{Since the index is 2, which is even; so the radicand is non-negative}\text{.}\\ x+4\ge 0\\ x\ge -\mathrm{4..............}[subtract\text{ 4 from each side]}\\ \text{Make a table of values for x}\ge -4\text{ and corresponding y-values}\text{.}\end{array}$

x	$y=-\frac{1}{3}\sqrt{x+4}-1$
-4	-1
-3	-1.33
-2	-1.47
-1	-1.58
0	-1.67
1	-1.75

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

  

  $\begin{array}{l}So,\text{ the domain of the function is {x | x}\ge -4\text{} and }\\ \text{the range of the function is {f(x) | f(x) }\le -1\text{ }}\text{.}\end{array}$