Chapter 7, Problem 32SGR

To determine

To Graph and Find :

Graph the given function and find the domain and range.

Answer to Problem 32SGR

  $\begin{array}{l}\text{The domain of the function is {x | x}\ge \text{0} and }\\ \text{the range of the function is {f(x) | f(x) }\le \text{0 }}\text{.}\end{array}$

Explanation of Solution

Given:

The function is:

  $f(x)=-\sqrt{6x}$

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:

  $\begin{array}{l}Given\\ f(x)=-\sqrt{6x}\\ \text{Since the index is 2, which is even; so the radicand is non-negative}\text{.}\\ 6x\ge 0\\ \frac{6x}{6}\ge \frac{0}{6}\\ x\ge 0\\ \text{Make a table of values for x}\ge \text{0 and corresponding y-values}\text{.}\end{array}$

x	$y=-\sqrt{6x}$
0	0
1	-2.449
2	-3.464
3	-4.242
4	-4.898
5	-5.477

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

  

  $\begin{array}{l}So,\text{ the domain of the function is {x | x}\ge \text{0} and }\\ \text{the range of the function is {f(x) | f(x) }\le \text{0 }}\text{.}\end{array}$