What would you call this?

Question

This is the functional equation of the form $$ f\left(\frac{x}{\sqrt{(x^2+(g(x))^2)}}\right) = \frac{g(x)}{\sqrt{(x^2+(g(x))^2)}} $$ we know that the solution for f of (x) is universally the function of unit circle and that makes g(x) to be any function, however if we put any analytical function in terms of those parameters we are not going to get full unit circle. "Parametric function" of the form $$ \left(\frac{t}{\sqrt{(t^2+(g(t))^2)}},\frac{g(t)}{\sqrt{(t^2+(g(t))^2)}}\right)$$ This is partial unit circle which has not equal density all through the circle. Is this parametric formation of function useful in any areas of mathematics. I will be thank full if any one can give me their thoughts on this form of not equally distributed not complete unit circle.

Answer 1

If your question is "what function $f$ solves the displayed equation", why isn't "$f(u)=\sqrt{1-u^2}$" an answer, or "any function $f$ for which $f(\cos x) = \sin x$ for all $x$" an answer?

But it is possible I misunderstood your question. If so, can you rephrase it please?