Question

4. Let \( f(x)=\frac{e^{x}-e^{-x}}{2} \) and if \( g(f(x))=x \), then \( \frac{1}{100}\left[g\left(\frac{e^{1002}-1}{2 e^{501}}\right)-1\right] \)

Answer 1

f(x) = \(\frac{e^x-e^{-x}}2\) = \(\frac{e^{2x}-1}{2e^x}\)

f(501) = \(\frac{e^{1002}-1}{2e^{501}}\) 

\(\therefore\) \(\frac1{100}[g(\frac{e^{1002}-1}{2e^{501}})-1]\) = \(\frac1{100}[g(f(501))-1]\)

= \(\frac1{100}(501-1)\) (\(\because\) g(f(x)) = x)

= 500/100 = 5