Domain, Range & Function Graphs
Text The domain of a function of two variables in polar coordinates is a subset of the polar coordinate plane { (r,θ)  r ∈ R \mbox{and} 0 {leq} θ < 2π.
The most common domains
are products of intervals of the form a {leq} r {leq} b and c {leq} θ {leq} d where 0 {leq} c {leq} d < 2π.
A more general polar coordinate domain is of the form r_{1}(θ{leq} r {leq} r_{2}(θ) and c {leq} θ {leq} d where 0 {leq} c {leq} d < 2π.and where 0 ≤ r_{1}θ) ≤r_{2}(θ) for two functions r_{1} and r_{2} of θ.
The range of a realvalued function f is the collection of all real numbers f(p) where p is in the domain of f.
The simplest example of a function is the constant function
that assigns the real number k to all p in the domain. The range of this
function is the set { k } containing one point.
The graph of a function of two variables is the
collection of points (x,y,f(x,y)) in 3space where (x,y) is in the domain of f. When we write the domain in polar coordinates, the graph is said to be in cylindrical coordinates.
Demos
Domain and Range
 
This demonstration graphs a function f[r,θ] over a disc domain. By default, the domain is set to 0 ≤ r ≤ 1, π ≤ θ ≤ π and the function is set to f[r,θ] = r^{2}cos(2θ). In the window labeled Domain and Range, you can choose both the center and radius of a red disc domain in the xyplane using the white and red hotspots respectively. The magenta line segment along the zaxis in this window shows the range of f[r.θ] over the red disc domain.

Exercises What is the range of the function f(x,y) = x^{4} + 2x^{2}  y^{2} over a unit disc domain centered at the origin (i.e. the set of all points (x,y) such that 0 ≤ r ≤ 1)?
Describe the graph of f(x,y) = 2xy/(x^{2} + y^{2}) for all values of x, y other than (0,0) where the function is not defined.
