How to perform vertical line test algebraically? I understand how a function should have one value mapped to a single value in the range. Thus, when graphing the function I understand how a vertical line can determine if a point or multiple points intersect the vertical line. This is visual. However, how can we perform the same test algebraically across a domain of the function?
I am learning calculus from the start again since college, so in case my question is naive or obvious I apologize.

A vertical line is the set of all points $(x_0,y)$ where $x_0$ is a fixed constant and $y$ is allowed to vary freely. Given an equation in the form $f(x,y)=0$, we say it passes the vertical line test if, for every fixed choice of $x_0$, the resulting equation $f(x_0,y)=0$ has either $0$ or $1$ solutions in terms of $y$.
For example, if we were checking the equation
$$ x = y^2 $$
we would define the function $f(x,y) = x - y^2$. If we choose $x_0=4$, notice that the equation
$$f(4,y) = 4 - y^2 = 0$$
fails the vertical line test since $y=2$ and $y=-2$ are both solutions.

As an exercise, see if you can prove that if $y$ is a function of $x$, such as $y=g(x)$, then the equation $f(x,y)=y-g(x)$ will always pass the vertical line test :)
