# Function of more than one variables

## Main Question or Discussion Point

Hi
I am confused a bit in functions.
Consider a function f(x)=x^2 . Putting various values of x here will give corresponding values of f(x) or y because y=f(x). OK
Now consider a function f(x,y)= x^2 + y^2 . Now putting here values of x and y will give value of f(x,y). Now what is f(x,y) here????? Above f(x) was y or distance from x axis. What is f(x,y)
here then. Is it z then?
If yes, then consider a function f(x,y,z) = x^2 + y^3 + z^2, then what is f(x,y,z) here?????????????

For f(x)=x^2, every x has a corresponding y, and the graph is a line, or curve.

For f(x,y)=x^2+y^2, every TWO numbers (x,y) have a corresponding z. Now you can see that just as all the x values are lined up in a one dimensional line, all the (x,y) values are contained in an AREA. And every point of that area has a corresponding z. (Yes, you were right, it's a z.) The graph then, is a SURFACE "hovering" above the area.

For f(x,y,z) = x^2 + y^3 + z^2 things get tricky. As we have seen, a function of one variable has a 2D graph (the simple graph you can draw in a piece of paper), and a function of two variables has a 3D graph (the surface I just discussed). So your question is, what about a function of three variables? It has a 4D graph? Well, theoretically, yes, but we humans can only visualize three dimensions, so the graph of these functions doesn't exist.

HallsofIvy
When you say "f(x) was y or distance above the x-axis" you are talking about the graph of the function. Of course, f(x) may have many different meanings depending on the particular application. If you wanted to graph z= f(x,y) you would need a three dimensional coordinate system with x, y, and z axes. In that case, z= f(x,y) would be "height above the xy-plane" and the graph would be a two dimensional surface. If you wanted to graph u= f(x,y,z) then you would need four perpendicular axes and that just isn't possible in three dimensional space. What we can do, often, is look at "level surfaces". Each value of f would give a surface. For example $f(x, y, z)= x^2+ y^2+ z^2= c$, for constant c, is a sphere with center at (0,0,0) and radius $\sqrt{z}$. Different values of f would give spheres with center at different radii.