What is the difference between f(x, y) and f(x, y, z)?

[makyol]

Homework Statement

What does f(x, y) function mean? Or What does f(x, y, z) function mean? Although, f(x, y) mean 3-dimensional, what is the matter with f(x, y, z)?

This is a general question, i am asking this question to really get the idea. Can anyone help me to figure out it?

Homework Equations

The Attempt at a Solution

 

[rock.freak667]

f(x) is a function in x only. You can draw this in the Cartesian plane in R²

f(x,y) is function in x and y. If you draw this in R³, the function will lie in the xy-plane.

f(x,y,z) is a function in x,y and z. In R³, the function lies in all three planes so to speak.

 

[Mark44]

f(x) is a function in x only. You can draw this in the Cartesian plane in R²

These explanations are somewhat misleading and somewhat incorrect. The graph of the equation y = f(x) is the set of ordered pairs (x, y) in R² where y = f(x). The domain of f is the entire x-axis or some subset of it.

f(x,y) is function in x and y. If you draw this in R³, the function will lie in the xy-plane.

The domain of the function is the x-y plane or some subset of it. The graph of the function is the ordered triples (x, y, z) for which z = f(x, y).

Example: z = ln(xy). The domain is the portion of the plane for which xy > 0, which is the interior of the first quadrant and the interior of the third quadrant.

f(x,y,z) is a function in x,y and z. In R³, the function lies in all three planes so to speak.

The domain of a function of three variables is R³ or a subset of it. The graph of w = f(x, y, z) is the set of ordered quadruples (x, y, z, w) such that w = f(x, y, z). Such a graph requires four dimensions: three for the domain and one for the range.

Example: w = x² + y² + z². Here the domain is all of R³, and the range is {w | w >= 0}.