Question

for each of the following equations, determine whether y is a function of x.
$x = 7y^2 + 5$ 〇 function 〇 not a function
$3x - y = 9$ 〇 function 〇 not a function
$y = 6x^2 - 2$ 〇 function 〇 not a function
$y^2 = 4x$ 〇 function 〇 not a function

Explanation:

Step1: Test $x=7y^2+5$

Rewrite to solve for $y$:
$7y^2 = x-5 \implies y^2 = \frac{x-5}{7} \implies y = \pm\sqrt{\frac{x-5}{7}}$
A single $x$ (e.g., $x=12$) gives two $y$-values ($y=1$ and $y=-1$), so it fails the vertical line test.

Step2: Test $3x-y=9$

Rewrite to solve for $y$:
$y = 3x - 9$
Each $x$ gives exactly one $y$-value, so it passes the vertical line test.

Step3: Test $y=6x^2-2$

This is a quadratic equation in $x$. Each $x$ input results in exactly one $y$-output, so it passes the vertical line test.

Step4: Test $y^2=4x$

Rewrite to solve for $y$:
$y = \pm\sqrt{4x} = \pm2\sqrt{x}$
A single $x$ (e.g., $x=1$) gives two $y$-values ($y=2$ and $y=-2$), so it fails the vertical line test.

Answer:

1. $x = 7y^2 + 5$: $\boldsymbol{\text{Not a function}}$
2. $3x - y = 9$: $\boldsymbol{\text{Function}}$
3. $y = 6x^2 - 2$: $\boldsymbol{\text{Function}}$
4. $y^2 = 4x$: $\boldsymbol{\text{Not a function}}$