You may also consider FunctionRange.

Thanks, Gianluca Gorni. Your explanation was helpful that the ReplaceAll[] in my ParametricPlot[] call was not causing ParametricPlot[] to use the correct domain. I assume from your answer that it is a limitation in Mathematica and not my misuse of ReplaceAll[].

What I am still uncertain about is whether there is some way to define a domain for ParametricPlot[] (and other plotting functions) that does not require foreknowledge of the expected image. If not, is there another plotting function that will accept a well-defined domain to a function.

Mathematica is great for function-defining. So a solution that includes preprocessing by other functions like Reduce[], Solve[], FunctionDomain[], etc. would answer my question.

Another related question is whether there is a way to know (or even guess) what singularities will be problematic for a plotting function. Plotting functions handle some singularities without even showing them.

Thanks again. Can you expand on that or point me to something that would explain it?

I understand why Annulus[] works in this example, but it doesn't generalize well since it requires more interpretation of the function than just identifying the singularity.

More importantly, I'm worried about some wrong assumptions I think I've been making all along. I didn't realize I was ignoring the domain. The example I posted uses ReplaceAll, but I get the same result using Assuming[], and Condition. Until now, I thought I was restricting my domain with those.

Also I don't understand the plotting problem with singularities. The plotting functions often handle singularities. In fact, they sometimes handle them too well since they can hide undefined points in rational expressions.

Yes, I meant the image of a function. And thanks, Gianluca Gorni. It looks like ParametricPlot[] might be the answer. This forum is great.

Your solution works for my example. However, my hope is for a more general solution that doesn't rely on foreknowledge of the image. In other words, I'm looking for a FunctionImage[function_] that takes a function and returns its image, similar to the way ImplicitRegion[] and Region[] work with equations. I tried,...

ParametricPlot[{x, y}/Sqrt[x^2 + y^2], {x, -2, 2}, {y, -2, 2},
  PlotRange -> {{-2, 2}, {-2, 2}}, Frame -> False, Axes -> True] /.
 Abs[x] -> PositiveReals || Abs[y] -> PositiveReals

The image I got was a bounded region as opposed to the closed curve. In other words, the image looks correct on the part of the domain 1<=Sqrt[x^2+y^2] but not on 0<Sqrt[x^2+y^2]<1

Do you know of a refinement that would work?

Another approach that almost worked was,...

Region[
  ImplicitRegion[{x, y}/Sqrt[x^2 + y^2] == {x, y}, {x, y}],
  Axes -> True, PlotRange -> {{-2, 2}, {-2, 2}}] /.
 Abs[x] -> PositiveReals || Abs[y] -> PositiveReals

The output here shows only two points of the image.