Q 10. - Calculus

Question

Let z = f(x, y) be a function of two variables. Explain why the two sets 

{z | (x, y, z) ∈ Graph( f )} and Range( f ) are identical.

Step-by-Step Solution

Answer

Sets {z | (x, y, z) ∈ Graph( f )} and Range( f ) are identical as both have one output variable.

Step 1. Given information

Sets are {z | (x, y, z) ∈ Graph( f )} and Range( f ) 

Step 2. Explanation

A graph is to be plotted between two variables, x, y and 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.

The output variable range of a function is the set of values.

There is one output variable for each point (x, y, z), the value of which determines the range of functions. z will be used to identify each output.

As a result, the sets {z | (x, y, z) ∈ Graph( f )} and Range( f )  are identical.