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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?

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2 Answers

up vote 4 down vote accepted

If $f(g(x))=x$, then $f$ and $g$ are inverses. Thus $g(f(x))=x$ as well. The answer is then $g(f(1006))=1006$.

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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

$$g(\frac{e^{2012}-1}{2e^{1006}}) = g\circ f(1006) = 1006$$

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Do you mean $1006$? –  Quixotic Jun 28 '11 at 14:46
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