A problem on function of a function

Question

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?

score 5 · Answer 1 · edited Jun 28 '11 at 14:49

5

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

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

edited Jun 28 '11 at 14:49

Apostolos

6,024

answered Jun 28 '11 at 14:41

jspecter

8,152

1

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

score 4 · Accepted Answer · answered Jun 28 '11 at 14:40

4

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

answered Jun 28 '11 at 14:40

J126

17,451

A problem on function of a function

2 Answers2