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.


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


$y^2=4ax$ 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?

Relevant threads:

  • $\begingroup$ 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? $\endgroup$ – Matti P. Jun 12 '18 at 12:22
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    $\begingroup$ To be precise, they mean that the equation y^2 = 4ax does not define a function that maps x to y. $\endgroup$ – Simon Jun 12 '18 at 13:41
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    $\begingroup$ You have it backwards. No single input can map to more than one output. $\endgroup$ – Phil H Jun 12 '18 at 13:47

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.


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