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Chapter 12: Problem 5

How many axes (or how many dimensions) are needed to graph the function \(z=f(x, y) ?\) Explain.

Short Answer

Expert verified
Answer: Three dimensions are required to graph the function \(z = f(x,y)\) as there are three variables involved: \(x\), \(y\), and \(z\).

Step by step solution

01

Determine the variables in the function

In the given function, \(z = f(x, y)\), there are three variables: \(x\), \(y\), and \(z\). The function takes \(x\) and \(y\) as input and returns \(z\) as output.
02

Visualize the axes corresponding to the variables

As there are three variables, we need three axes to represent them, one for each variable. The standard convention for axis labelling is to use \(x\)-axis for the \(x\) variable, \(y\)-axis for the \(y\) variable, and \(z\)-axis for the \(z\) variable.
03

Explain the need for three dimensions

In order to properly visualize the function \(z = f(x,y)\), we require three dimensions because there are three variables. In a three-dimensional coordinate system, each point \((x, y, z)\) denotes a unique position where the \(z\)-value corresponds to the function value \(f(x,y)\) for each given pair \((x,y)\). This enables us to fully appreciate the relationship between the input variables \((x,y)\) and the output variable \(z\).

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