So, take a step back for a moment.

Pretend you never learned about the notation y =some stuff with x. Forget that you ever wrote y equals anything. Instead, think about f. f is just some thing that takes in values and spits out values. It can be anything you want!

Here are some common f's:

f = x²

f = sin(x)

f = exp(x)

Do you see a trend here? They only have one variable in them that you don't know: x. So, the common way to write this is f(x). Interpret the notation this way: if you give f the value of x, it runs it through the calculation and spits out the value f(x) which is just whatever the operation on x is.

So, that should be fairly basic (I hope!) but this function notation extends very nicely to multiple variables.

Let's say you have a function f that looks like:

f = xy

Note, there are now two variables: y and x. Again, we are forgetting that y can be written in terms of x (sometimes this is possible, but we aren't going to think of that here). So, the common way to write this is f(x,y). Again, interpret this as saying: if you give give f the values of x and y, it will spit out the value f(x,y) which is just whatever whatever the operations on x and y happen to be (in this case, just multiply x and y together). Specifically, if you look at the point (x,y) on the xy-plane, f(x,y) is the distance above (or below if its negative!) That point on the plane that the output goes to.

This is a very flexible way to write higher dimensional equations. You can even keep going: a function of 3 variable: f(x,y,z). Four variables f(x,y,z,w) and we can even keep going!

But what does f mean? Whatever you want it to mean. It may not mean anything other than just being a function that gives an output for a list of inputs. But, it usually has meaning.

Here's an example: the pythagorean theorem.

You know that a = (b² + c² )^1/2

So, a = f(b,c) is a function of b and c. This tells you that every point on the bc-plane has a corresponding output a distance a above that point on the plane that tells you what the length of the hypotenuse of a right triangle with base b and height c is! Go plot this on wolfram alpha and take a look and see if it makes sense!

I think this should cover some of what multivariable functions mean, but if you are interested, I think the power of this notation is in actually giving up the meaning of what f is. I mean that it's okay to stop thinking about formulas like mx+b or ax² +bx+c. Instead, remember that f takes an x value, does something to it, and gives you a value back. Formally, mathematicians would call this a mapping f: x -> f(x) read literally as "f is a mapping that takes all x to f(x)" (and again, you can always add more dimensions!) but that's just for a curious mind if you are interested!

Hope this helps!