# Find the function that is implicitly defined in the relation

We are given a relation $$\sqrt{ x^{2} - y^{2}} + \cos^{-1} (\frac{x}{y}) = 0$$ and are asked to find a function $$g(x)$$ that is implicitly defined within it. $$y = x$$ works but how can I show this is the only function that works and if there are more how can I find them?

• Are you searching for a function $h(x,y)$ or for a function $g(x)$ since your title and your text body are stating different things :) – mrtaurho Oct 4 '18 at 16:47
• Thanks for that, I changed the wording in the question and forgot to change the title accordingly. – Diehardwalnut Oct 4 '18 at 16:59

The main branches of $$\sqrt{\cdot}$$ and $$\cos^{-1}$$ have nonnegative ranges, so their sum is zero only if they are both zero themselves. This forces $$y = x$$.