Question

for each of the following, determine whether the equation defines y as a function of x.
$2x + |y| = 0$ $circ$ function $circ$ not a function
$y = \sqrt{x + 5}$ $circ$ function $circ$ not a function
$x^2 + 9y = 4$ $circ$ function $circ$ not a function
$x^2 + y^2 = 4$ $circ$ function $circ$ not a function

Explanation:

Step1: Test $2x+|y|=0$ for function

Rewrite: $|y|=-2x$, so $y=\pm(-2x)$. For a single valid $x>0$, there are 2 $y$-values. This fails the vertical line test.

Step2: Test $y=\sqrt{x+5}$ for function

For each valid $x\geq-5$, there is exactly one non-negative $y$-value. Passes vertical line test.

Step3: Test $x^2+9y=4$ for function

Rewrite: $y=\frac{4-x^2}{9}$. For every $x$, there is exactly one $y$-value. Passes vertical line test.

Step4: Test $x^2+y^2=4$ for function

Rewrite: $y=\pm\sqrt{4-x^2}$. For a single valid $x$ (e.g., $x=0$), there are 2 $y$-values. Fails vertical line test.

Answer:

1. $2x + |y| = 0$: $\boldsymbol{\text{Not a function}}$
2. $y = \sqrt{x + 5}$: $\boldsymbol{\text{Function}}$
3. $x^2 + 9y = 4$: $\boldsymbol{\text{Function}}$
4. $x^2 + y^2 = 4$: $\boldsymbol{\text{Not a function}}$