1. **Problem 1a: Analyze the equation $|y| = x$.** - The equation states that the absolute value of $y$ equals $x$. - This means $y$ can be either $x$ or $-x$ when $x \geq 0$, but no real $y$ exists for $x < 0$ because absolute value is always non-negative. 2. **Check if $y$ is a function of $x$.** - For $x \geq 0$, there are two possible $y$ values: $y = x$ and $y = -x$. - A function must assign exactly one output $y$ for each input $x$. - Here, each $x \geq 0$ corresponds to two $y$ values, so it is not a function. 3. **Problem 1b: Analyze the equation $y^2 = x^2$.** - This implies $y = \pm x$. - For each $x$, there are two possible $y$ values: $y = x$ and $y = -x$. 4. **Check if $y$ is a function of $x$.** - Since each $x$ corresponds to two $y$ values, this relation is not a function. **Summary:** Both equations fail the vertical line test because for some $x$ values, there are two corresponding $y$ values. $$\boxed{\text{Neither } |y|=x \text{ nor } y^2=x^2 \text{ defines } y \text{ as a function of } x.}$$