Chapter 1.5, Problem 41E

Solution

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

The ordered pair which is the inverse of the given relation is option $E.\left(1,-2\right)$

Given:

Given the function: $x^{2}y+5y=9$

Options:

  $\begin{array}{l}A.\left(2,1\right)\hfill \\ B.\left(-2,1\right)\hfill \\ C.\left(-1,2\right)\hfill \\ D.\left(2,-1\right)\hfill \\ E.\left(1,-2\right)\hfill \end{array}$

Calculation:

The inverse of the function $x^{2}y+5y=9$ is $y^{2}x+5x=9$ which is obtained by switching the $x$ and the $y$ variables.

Check for option $\left(A\right)\text{. }\left(2,1\right)$ :

Substitute $2$ for $x$ and $1$ for $y$ , in the inverse relation $y^{2}x+5x=9$ :

  $\begin{array}{c}{\left(1\right)}^2\left(2\right)+5\left(2\right)=9\\ 2+10=9\\ 12\ne 9\end{array}$

The equation is not satisfied. So, option $\left(A\right)\text{. }\left(2,1\right)$ is not in the inverse relation.

Check for option $B.\left(-2,1\right)$ :

Substitute $-2$ for $x$ and $1$ for $y$ , in the inverse relation $y^{2}x+5x=9$ :

  $\begin{array}{c}{\left(1\right)}^2\left(-2\right)+5\left(-2\right)=9\\ -2-10=9\\ -12\ne 9\end{array}$

The equation is not satisfied. So, option $B.\left(-2,1\right)$ is not in the inverse relation.

Check for option $C.\left(-1,2\right)$ :

Substitute $-1$ for $x$ and $2$ for $y$ , in the inverse relation $y^{2}x+5x=9$ :

  $\begin{array}{c}{\left(2\right)}^2\left(-1\right)+5\left(-1\right)=9\\ -4-5=9\\ -9\ne 9\end{array}$

The equation is not satisfied. So, option $C.\left(-1,2\right)$ is not in the inverse relation.

Check for option $D.\left(2,-1\right)$ :

Substitute $2$ for $x$ and $-1$ for $y$ , in the inverse relation $y^{2}x+5x=9$ :

  $\begin{array}{c}{\left(-1\right)}^2\left(2\right)+5\left(2\right)=9\\ 2+10=9\\ 12\ne 9\end{array}$

The equation is not satisfied. So, option $D.\left(2,-1\right)$ is not in the inverse relation.

Check for option $E.\left(1,-2\right)$ :

Substitute $1$ for $x$ and $-2$ for $y$ , in the inverse relation $y^{2}x+5x=9$ :

  $\begin{array}{c}{\left(-2\right)}^2\left(1\right)+5\left(1\right)=9\\ 4+5=9\\ 9=9\end{array}$

The equation is satisfied. Thus, option $E.\left(1,-2\right)$ is in the inverse relation.