The discussion clarifies the differences between the functions f(x, y) and f(x, y, z) in the context of dimensionality and graph representation. f(x, y) represents a function in two dimensions, specifically lying in the xy-plane of R3, while f(x, y, z) extends this concept to three dimensions, requiring a four-dimensional representation for its graph. The domain of f(x, y) is a subset of the xy-plane, whereas the domain of f(x, y, z) encompasses R3 or its subsets. Examples provided include z = ln(xy) for f(x, y) and w = x² + y² + z² for f(x, y, z).
PREREQUISITESStudents studying calculus, mathematicians interested in multivariable functions, and educators teaching concepts of dimensionality and graphing in mathematics.
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.rock.freak667 said:f(x) is a function in x only. You can draw this in the Cartesian plane in R2
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).rock.freak667 said: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 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.rock.freak667 said:f(x,y,z) is a function in x,y and z. In R3, the function lies in all three planes so to speak.