Question

for each of the following equations, determine whether y is a function of x.
$y^2 = 5x - 9$ \t\t○ function ○ not a function
$x = y^2$ \t\t○ function ○ not a function
$y = \frac{1}{2}x$ \t\t○ function ○ not a function
$3x + 6y = 12$ \t○ function ○ not a function

Explanation:

Step1: Apply vertical line test to $y^2=5x-9$

Solve for $y$: $y=\pm\sqrt{5x-9}$. One $x$ gives two $y$-values.

Step2: Apply vertical line test to $x=y^2$

Solve for $y$: $y=\pm\sqrt{x}$. One $x$ gives two $y$-values.

Step3: Apply vertical line test to $y=\frac{1}{2}x$

Linear form: one $x$ gives one $y$-value.

Step4: Simplify $3x+6y=12$ and test

Rewrite as $y=-\frac{1}{2}x+2$. One $x$ gives one $y$-value.

Answer:

1. $y^2 = 5x - 9$: $\boldsymbol{\text{Not a function}}$
2. $x = y^2$: $\boldsymbol{\text{Not a function}}$
3. $y = \frac{1}{2}x$: $\boldsymbol{\text{Function}}$
4. $3x + 6y = 12$: $\boldsymbol{\text{Function}}$