I am aware of the vertical line test. If you place a vertical line over a shape, and if it crosses more than once, it fails the vertical line test and is no longer a function.
But I don't understand why? To me a function is just input and output value:
$$f(x)=x^2$$
You enter an input value and you get an output value. Above is a parabola, since it is a basic quadratic function.
But you can do the same with hyperbolas:
$$f(x,y)={{x^2}\over9}–{{y^2}\over4}$$
And you can do the same with ellipses, which are essentially the inverse of a hyperbola:
$$f(x,y)={{x^2}\over9}+{{y^2}\over4}$$
To me, those two are functions, as is the original quadratic. True, the latter two fail the vertical line test, but so what? You give it an input value and you get an output value. Why are these not functions?
Best Answer
The function $f(x)=x^2$ is not a parabola. It is merely a function.
If you write $y=x^2,$ then the set of points in the $x,y$-plane that satisfy that equation is a parabola.
The "vertical line" test in a Cartesian plane tells you precisely whether a figure in the plane is a graph of a function of exactly one variable, plotted using the horizontal axis (usually $x$) to plot the input of the function and the vertical axis (usually $y$) to plot the output of the function.
When you write $f(x,y)=\frac{x^2}9 - \frac{y^2}4,$ then certainly $f(x,y)$ is a function, but it's a function of two variables, and you are not plotting its output using the vertical axis. In fact, that equation does not even define any kind of curve in the $x,y$-plane, because you've put no restrictions on what $x$ or $y$ could be. Any values of $x$ and $y$ are valid input for this $f(x,y)$ and will make the equation true, since you have defined the function that way.
If you write $$\frac{x^2}9 - \frac{y^2}4 = 0,$$ then you have written an equation describing a curve in the $x,y$-plane. That curve is still not the graph of a function of $x$ whose output value is $y,$ and the vertical line test confirms that it is not.
Edit: I substituted new wording in a couple of places that I hope is a little more accurate than my previous use of the words "valid" and "relevant." Moreover, I've attempted to make it clearer that the comments above apply only to figures in a plane. You can define a figure in three-dimensional Cartesian space by the points that satisfy $z=f(x,y)$ for some function $f$ of two variables, and this figure will pass a "vertical line test" where the "vertical" lines are parallel to the $z$-axis. Thanks go to the commenters who pointed out these weaknesses in the original answer text.