Homeomorphism from $S^1\setminus(0,1)$ to $\mathbb{R}$

I'm sorry to say I don't have any great ides for solving your equation for $f(x)$. You could try getting the radical that involves $f(x)$ by itself and squaring both sides but it'll be pretty awful, I would assume. Thankfully there is a significantly easier approach utilizing more geometry, less algebra.

Small triangle II, with vertices $(0,1),\, (w, h)$, and $(0,h)$, is similar to the "overall" triangle with vertices $(0, 1),\, (f(x), 0)$, and $(0, 0)$.

Thus the ratios of heights and widths are the same; $$\frac{h}{1} = \frac{w}{f(x)},$$

and you can easily solve for $f(x)$.