Question

In the following exercises, determine whether each equation is a function.

a. $2x-4y=8$$2x-4y=8$

b. $-4=x^2-y$$-4=x^2-y$

c.$y^2=-x+5$

Short answer

The equations$2x-4y=8$and$-4=x^2-y$are functions.

Step-by-step solution

Step 1. Given Information

The given equations are:

$$\begin{gathered} 2x-4y=8 \\ -4=x^2-y \\ {y}^2=-x+5 \end{gathered}$$

Part (a) Step 1. To check if 2x-4y=8 is a function.

For a given value of $x,$we multiply it by -2 and then add $8$to it. We then divide this sum by $-4.$

role="math" localid="1644670215195" $y=\frac{-2x+8}{-4}$

For example, when $x=2$:

role="math" localid="1644670243142" $$\begin{gathered} y=\frac{-2\left(2\right)+8}{-4} \\ y=\frac{-4+8}{-4} \\ y=\frac{4}{-4} \\ y=-1 \end{gathered}$$

We have, when $x=2,$then $y=-1$. It works similarly for all values of $x.$Since each value of $x$corresponds to only one value of $y$, the given equation defines a function.

Part (b) Step 1. To check if -4=x2-y is a function

For a given value of $x,$we square it and add it to $4$to get the value of$y.$

$y=x^2+4$

For example, when$x=1,$

$$\begin{gathered} y=1^2+4 \\ y=1+4 \\ y=5 \end{gathered}$$

We have that when $x=1,$then $y=5$.Since each value of $x,$corresponds to only one value of $y,$the given equation is a function.

Part (c) Step 1. To check if y2=-x+5 is a function.

When $x=-4$,

$$\begin{gathered} y^2=-(-4)+5 \\ {y}^2=9 \end{gathered}$$

Here we have solutions: $y=3$and$y=-3.$

We have shown that when $x=-1,$then $y=3$and $y=-3.$It works similarly for all values of $x.$

Since each value of$x$does not correspond to only one value of$y,$the given equation is not a function.

Step 5. Conclusion

Therefore, only the equations $2x-4y=8$and$-4=x^2-y$are functions,