Understanding the vertical line test of functions

I am reading a book about Calculus and in the first chapter it's said there's a test to validate whether a graph is a graph of a function or not. If it's possible to draw a vertical line that touches two points of the graph the graph is not the graph of a function. In other words, no two inputs can map to the same output.

So:

$f(x)=4x$ is a function

but

$y^2=4ax$ is not

However, to me that doesn't seem to make much sense as we could simply define a bivariate function like $f(x, y)$ as having two inputs instead.

Is there an intuitive way to understand this?

• So you're saying that $y^2 = 4ax$ is not a function (correct) and you could define a bivariate function, how would you define it in this case? – Matti P. Jun 12 '18 at 12:22
Any equation with variables $x$ and $y$ can be rearranged to $g (x,y)=0$ for some suitable multivariate function $g$.
But "the collection of zeros of some function" is not "the graph of a function". The word "graph" has a specific meaning here: the graph means the set of pairs $(x,f (x))$ (or for a multivariate function, it would be things like the surface that's the set of all triples $(x,y,g(x,y))$).
The set of pairs satisfying $y^2=x$ can't be the set of pairs $(x,f (x))$ no matter how hard you try to find a function $f$, because $f(1)$ would have to be both $1$ and $-1$. This sort of issue is observed geometrically as the vertical line $x=1$ intersecting the curve at two spots, hence the vertical line test.