Chapter 1.4, Problem 84E

To determine

To find:which of the functions $f\left(x\right)=cx$ , $g\left(x\right)=c{x}^2$ , $h\left(x\right)=c\sqrt{\left|x\right|}$ and $r\left(x\right)=\frac{c}{x}$ can be used to model the given data and also find the value of the constant c that will make the function fit the data in the table.

Answer to Problem 84E

  $h\left(x\right)=c\sqrt{\left|x\right|}$

  $c=3$

Explanation of Solution

Given:

The data in the table below:

x	-4	-1	0	1	4
y	6	3	0	3	6

Observing the function it can be function it can be said that the function value is decreasing as the value of x increases.

So, there are two options for the function $r\left(x\right)=\frac{c}{x}$ and $h\left(x\right)=c\sqrt{\left|x\right|}$ .

But since the function is defined for 0, so the possibility of the function being $r\left(x\right)=\frac{c}{x}$ is eliminated.

Thus, $h\left(x\right)=c\sqrt{\left|x\right|}$ is the function that can be used to model the given data.

Now, to find the value of c , substitute any one pair of values in the function $h\left(x\right)=c\sqrt{\left|x\right|}$ and evaluate it to find c .

So, let’s substitute $x=1$ , then by the above table $h\left(1\right)=3$ .

That is,

  $\begin{array}{c}h\left(1\right)=c\sqrt{\left|1\right|}\\ 3=c\sqrt{1}\\ 3=c\end{array}$

Conclusion:

The required function is $h\left(x\right)=3\sqrt{\left|x\right|}$ .