Q 9. - Calculus

Question

Let z = f(x, y) be a function of two variables. Explain why the two sets {(x, y) | (x, y,z) ∈ Graph( f )} and Domain( f ) are identical.

Step-by-Step Solution

Answer

The two sets {(x, y) | (x, y, z) ∈ Graph( f )} and Domain( f ) are identical as both have two input variables.

Step 1. Given information

Two sets are {(x, y) | (x, y, z) ∈ Graph( f )} and Domain( f ). 

Step 2. Explanation

The graph must be plotted between two variables, '$x,y,z$', for the function z=f(x, y).

As a result, there will be three axes to plot on the graph. This implies that f's graph must be a subset of $R^2$. (x, y, z) will be a subset of each point on the graph.

A function's domain is the set of input variable values.

There are two input variables in each point (x, y, z), whose values determine the domain set of functions. (x, y) will be used to represent each input.

As a result, the Domain (f) set and the set $\left\{\left(x,y\right)|\left(x,y,z\right)\in Graph\left(f\right)\right\}$ are identical.