[Math] graph of complicated equation

Question

Graph of the equation $(x+y) (x^2 + y^2 -1) = 0$ is just the line $y=-x$ and the circle $x^2 + y^2 = 1$.

Is it generally true that the graph of $f(x,y) \cdot g(x,y) = 0$ may be drawn as union of graphs of $f(x,y) =0 $ and $g(x,y) = 0$?

Answer 1

Note that the equation you've written is an equation, and not a function. And the graph is the union of the equations $$y = -x \;\; \cup\;\; x^2 + y^2 = 1$$

Yes indeed: $$f(x, y)\cdot g(x,y) = 0 \iff f(x, y) = 0 \;\;\cup\;\; g(x, y) = 0$$