Chapter 6, Problem 1SE

To determine

To choose: The correct option for an equation in which $y$ can be written as a function of $x$.

Answer to Problem 1SE

The correct answer from the given choices is $\underset{\_}{3x+2y=6}$.

Explanation of Solution

Given:

The equations for the choices A, B, C and D are as follows.

$\begin{array}{l}\left(\text{A}\right)3x+2y=6\\ \left(\text{B}\right)x\text{=}\sqrt{\left|y\right|}\\ \left(\text{C}\right)x\text{=}\left|y+3\right|\\ \left(\text{D}\right)x^2+y^2=9\end{array}$

Definition used:

“For $y$ to be a function of $x$, it is necessary that every input value of $x$ in the domain leads to one and only one value of $y$.”

Calculation:

Check the choices for the equation to be function of $x$ as follows.

Simplify the equation given in choice (A) as below:

$\begin{array}{l}3x+2y=6\\ 2y=-3x+6\left(\text{Subtract }3x\text{and}\text{rewrite}\right)\\ y=-\frac{3}{2}x+6\left(\text{Divide}\text{by}2\right)\end{array}$

In the above expression $y=-\frac{3}{2}x+6$, $y$ is written as a function of $x$ as for every value of $x$ in the domain leads to one and only one value of $y$.

Therefore, $y$ can be written as a function of $x$.

Simplify the equation given in choice (B) as below.

$\begin{array}{l}x=\sqrt{\left|y\right|}\\ x=\sqrt{y}\text{ or }x=\sqrt{-y}\left(y=\left|x\right|\Rightarrow y=x\text{ or }y=-x\right)\end{array}$

In the above expression, for one value of $x$ in the domain leads to more than one values of $y$.

Therefore, $y$ cannot be written as a function of $x$.

Simplify the equation given in choice (C) as below.

$\begin{array}{l}x=\left|y+3\right|\\ x=y+3\text{ or }x=-y-3\left(y=\left|x\right|\Rightarrow y=x\text{ or }y=-x\right)\end{array}$

In the above expression, for one value of $x$ in the domain leads to more than one values of $y$.

Therefore, $y$ cannot be written as a function of $x$.

Simplify the equation given in choice (D) as below:

$\begin{array}{l}{x}^2+y^2=9\\ y^2=9-x^2\left(\text{Subtract }{x}^2\right)\\ y=\pm \sqrt{9-x^2}\left(\text{Square root}\right)\end{array}$

In the above expression, for every value of $x$ in the domain leads to more than one values of $y$.

Therefore, $y$ cannot be written as a function of $x$.

Thus, the correct answer from the given choices is $\underset{\_}{3x+2y=6}$.