Function of more than one variables

[R Power]

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?

 

[LucasGB]

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 \(\displaystyle f(x, y, z)= x^2+ y^2+ z^2= c\), for constant c, is a sphere with center at (0,0,0) and radius \(\displaystyle \sqrt{z}\). Different values of f would give spheres with center at different radii.

But you should not think of f(x), or f(x,y), or f(x,y,z) as necessarily having anything to do with distances or heights. They might be the temperature at a particular point on the line, or plane, or space, respectively. Or they might be pressure, or magnetic strength or whatever.