Chapter 1.4, Problem 87E

To determine

Find the matching and determining Constants.

Answer to Problem 87E

The value of $c$ for $r\left(x\right)is\boxed{32}$ .

Explanation of Solution

Given information:

Match the data with one of the following functions and determine the value of the constant that will make the function fit the data in the table

  $f\left(x\right)=cx$ , $g\left(x\right)=c{x}^2$ , $h\left(x\right)=c\sqrt{\left|x\right|}$ and $r\left(x\right)=\frac{c}{x}$

  

Calculation:

Consider the given table.

  

Evaluate the given functions to satisfy the above data,

  $f\left(x\right)=cx$ , $g\left(x\right)=c{x}^2$ , $h\left(x\right)=c\sqrt{\left|x\right|}$ and $r\left(x\right)=\frac{c}{x}$

Consider $f\left(x\right)=cx$

  $\begin{array}{l}f\left(-4\right)=c\left(-4\right)\\ =-4c\end{array}$

This function can be equal to $-8$ if $c=2$

  $\begin{array}{l}f\left(1\right)=c\left(1\right)\\ =c\end{array}$

This function can be equal to $32$ if $c=32$

There is contradiction,

Hence the function cannot be matched with the data.

Now consider $g\left(x\right)=c{x}^2$

  $\begin{array}{l}g\left(-4\right)=c{\left(-4\right)}^2\\ =16c\end{array}$

This function can be equal to $-8$ if $c=-\frac{1}{2}$

  $\begin{array}{l}g\left(-1\right)=c{\left(-1\right)}^2\\ =c\end{array}$

This function can be equal to $-32$ if $c=-32$

Hence the function cannot be matched with the data.

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

  $\begin{array}{l}h\left(-4\right)=c\sqrt{\left|-4\right|}\\ =c\sqrt{4}\\ =2c\end{array}$

This function can be equal to $-8$ if $c=-4$

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

This function can be equal to $32$ if $c=-32$

This function cannot be matched with the data.

Now consider $r\left(x\right)=\frac{c}{x}$

  $r\left(-4\right)=\frac{c}{-4}$

This function can be equal to $-8$ if $c=32$

  $\begin{array}{l}r\left(1\right)=\frac{c}{1}\\ =c\end{array}$

This function can be equal to $32$ if $c=32$

We get the required function.

This function matched with the data.

Hence, the value of $c$ for $r\left(x\right)is\boxed{32}$ .