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?

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    $\begingroup$ 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 :) $\endgroup$ – mrtaurho Oct 4 '18 at 16:47
  • $\begingroup$ Thanks for that, I changed the wording in the question and forgot to change the title accordingly. $\endgroup$ – 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 $y = x$.


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