Chapter 1.5, Problem 42E

Solution

To Determine: The ordered pair which is the inverse of the given relation.

The ordered pairs which are the inverse of the given relations are:

Option $A.\left(0,-4\right)$

Option $C.\left(3,2\right)$

Option $D.\left(2,12\right)$

option $E.\left(1,-6\right)$

Given:

Given the function: $x{y}^2-3x=12$

Options:

  $\begin{array}{c}A.\left(0,-4\right)\\ B.\left(4,1\right)\\ C.\left(3,2\right)\\ D.\left(2,12\right)\\ E.\left(1,-6\right)\end{array}$

Calculation:

The inverse of the function $x{y}^2-3x=12$ is $y{x}^2-3y=12$ which is obtained by switching the $x$ and the $y$ variables.

Check for option $A.\left(0,-4\right)$ :

Substitute $0$ for $x$ and $-4$ for $y$ , in the inverse relation $y{x}^2-3y=12$ :

  $\begin{array}{c}\left(-4\right){\left(0\right)}^2-3\left(-4\right)=12\\ 0+12=12\\ 12=12\end{array}$

The equation is satisfied. So, option $A.\left(0,-4\right)$ is in the inverse relation.

Check for option $B.\left(4,1\right)$ :

Substitute $4$ for $x$ and $1$ for $y$ , in the inverse relation $y{x}^2-3y=12$ :

  $\begin{array}{c}\left(1\right){\left(4\right)}^2-3\left(1\right)=12\\ 16-3=12\\ 13=12\end{array}$

The equation is not satisfied. So, option $B.\left(4,1\right)$ is not in the inverse relation.

Check for option $C.\left(3,2\right)$ :

Substitute $3$ for $x$ and $2$ for $y$ , in the inverse relation $y{x}^2-3y=12$ :

  $\begin{array}{c}\left(2\right){\left(3\right)}^2-3\left(2\right)=12\\ 18-6=12\\ 12=12\end{array}$

The equation is satisfied. So, option $C.\left(3,2\right)$ is in the inverse relation.

Check for option $D.\left(2,12\right)$ :

Substitute $2$ for $x$ and $12$ for $y$ , in the inverse relation $y{x}^2-3y=12$ :

  $\begin{array}{c}\left(12\right){\left(2\right)}^2-3\left(12\right)=12\\ 48-36=12\\ 12=12\end{array}$

The equation is satisfied. So, option $D.\left(2,12\right)$ is in the inverse relation.

Check for option $E.\left(1,-6\right)$ :

Substitute $1$ for $x$ and $-6$ for $y$ , in the inverse relation $y{x}^2-3y=12$ :

  $\begin{array}{c}\left(-6\right){\left(1\right)}^2-3\left(-6\right)=12\\ -6+18=12\\ 12=12\end{array}$

The equation is satisfied. So, option $E.\left(1,-6\right)$ is in the inverse relation.