Question

Graph the functions \(\begin{aligned}{l}f(x,y) = \sqrt {{x^2} + {y^2}} ,f(x,y) = {e^{\sqrt {{x^2} + {y^2}} }}\\f(x,y) = ln\sqrt {{x^2} + {y^2}} ,f(x,y) = sin\left( {\sqrt {{x^2} + {y^2}} } \right)\end{aligned}\) and\(f(x,y) = \frac{1}{{\sqrt {{x^2} + {y^2}} }}\). In general, if g is a function of one variable, how is the graph of \(f(x,y) = g(\sqrt {{x^2} + {y^2}} )\)obtained from the graph of g?

Short answer

The graph of the function \(f(x,y) = g(\sqrt {{x^2} + {y^2}} )\) is obtained by rotating the graph of \(z = g(x)\)about the Z- axis.

Step-by-step solution

Graph the functions   

\(f(x,y) = \sqrt {{x^2} + {y^2}} ,f(x,y) = {e^{\sqrt {{x^2} + {y^2}} }},f(x,y) = \ln \sqrt {{x^2} + {y^2}} \),

Consider the function\(f(x,y) = \sqrt {{x^2} + {y^2}} \)

The graph of the given function is given by:

Consider the function\(f(x,y) = {e^{\sqrt {{x^2} + {y^2}} }}\)

The graph of the given function is given by:

Consider the function\(f(x,y) = \ln \sqrt {{x^2} + {y^2}} \)

The graph of the given function is given by:

Graph the functions   

\(f(x,y) = \sin \left( {\sqrt {{x^2} + {y^2}} } \right),f(x,y) = \frac{1}{{\sqrt {{x^2} + {y^2}} }}\),

Consider the function\(f(x,y) = \sin \left( {\sqrt {{x^2} + {y^2}} } \right)\)

The graph of the given function is given by:

Consider the function\(f(x,y) = \frac{1}{{\sqrt {{x^2} + {y^2}} }}\)

The graph of the given function is given by:

Consider a function\(f(x,y) = g(\sqrt {{x^2} + {y^2}} )\).

Let g is a function of one variable.

To graph the function\(z = f(x,y) = g(\sqrt {{x^2} + {y^2}} )\), rotate the graph of\(z = g(x)\)about the Z- axis.

Therefore, the graph of the function \(f(x,y) = g(\sqrt {{x^2} + {y^2}} )\)is obtained by rotating the graph of\(z = g(x)\)about the Z- axis.