TITLE: A problem on function of a function
QUESTION [5 upvotes]: I am stuck with this problem,

A function $f(x)$ is defined as $f(x)
 = \sinh(x)$. Another function $g(x)$ is such that $f(g(x)) = x$.
Find the value of $\large
 g(\frac{e^{2012}-1}{2e^{1006}})$

I tried representing $f(x) =\large \frac{e^{2x}-1}{2e^{x}}$,and then using some algebraic manipulation but I am unable to get the correct answer. Any ideas?

REPLY [5 votes]: If $f(g(x))=x$, then $f$ and $g$ are inverses.  Thus $g(f(x))=x$ as well.  The answer is then $g(f(1006))=1006$.

REPLY [5 votes]: The idea is to notice $\mathrm{sinh}:\mathbb{R} \rightarrow \mathbb{R}$ is a bijection. Hence, any right inverse is a left inverse. It follows 
$$g(\frac{e^{2012}-1}{2e^{1006}}) = g\circ f(1006) = 1006$$