Can any non-piecewise rectangular function $y=f(x)$ be written as a non-piecewise polar function $r=g(θ)$?
I tried exploring a few examples—such as converting $y=\sin x$ into polar form—but most of these are too difficult for me as they involve big-O notation.
I am just looking for a yes or no and a brief explanation or example that shows why or how the answer is yes or no.
No.
Consider $f(x) = x$. In polar form, this has the equation $\theta = \pi/4$. There's no way to write this in the form of a polar function $r = g(\theta)$.