Behaviour of multivariate functions

Question

I'm trying to understand how multivariate functions work a little bit. For example, consider this function:

$$ f(x,\ y,\ z) = x^3 - 10x + y - z + 3$$

I can't graph this function on a 3D online graphing calculator, yet something like this works perfectly well:

$$f(x,\ y) = (x-2)^2 + (y-2)^2$$

$(x-2)^2 + (y-2)^2$ can be represented with the following graphic that I got from GeoGebra:

How come this one is capable of being graphed and not the first function? Is that because there are too many variables to deal with? Would it need a 4D calculator? Why is the second function multivariate if it has no $z$ quantity in it? My guess is that $(x-2)^2 + (y-2)^2$ equals $z$ but I don't know how I can assume that, it's just an inference I made from the graph. Regardless, if the second function is a multivariate function, then how do we distinguish it from the first one? And how does this explain why the first cannot be graphed with a 3D calculator?

Answer 1

Yes, your first function has three independent variables and one dependent variable, so you would not be able to graph in in a $3D$ graphing calculator. As for why the $(x-2)^{2} + (y-2)^{2}$ equals $z$, it's the same idea as in two dimensional graphing when we often interchange the notations $$f(x) = x^{2} \quad \text{and}\quad y = x^{2}.$$ Here instead of replacing $f(x)$ with $y$ we are replacing $f(x,y)$ with $z$. You could do something similar with $f(x, y, z) = x^3 - 10x + y - z + 3$ by introducing a new variable, say $w$, and writing it as $w = x^3 - 10x + y - z + 3.$

Answer 2

This has to do with human visuals.

We can PERCEIVE things in three dimensions, so it naturally makes sense to set $y=f(x)$, or $z=f(x,y)$, because these are graph able, TANGIBLE.

But when it comes to $w=f(x,y,z)$, there isn't a way to graph it, per se.

Some methods of graphing are used commonly, such as a color coded graph. For each point (x,y,z) on the domain of $f$, there will be a value associated with it ==> turned into a color through a conversion table.

For example, say you have a spherical lamp, and you want to find the temperature of the surface of the lamp. Let the temperature be $H=f(x,y,z,)$, defined on the sphere $x^2+y^2+z^2=1$.

Then, your graph would look like a thermal photo of a lamp.

So, it is not the same type of graph as $f(x,y)$ or $f(x)$, but it is still representable.