Functions Question 13
Question: The inverse of the function $ f(x)=\frac{e^{x}-{e^{-x}}}{e^{x}+{e^{-x}}}+2 $ is given by
[Kurukshetra CEE 1996]
Options:
A) $ {\log_{e}}{{( \frac{x-2}{x-1} )}^{1/2}} $
B) $ {\log_{e}}{{( \frac{x-1}{3-x} )}^{1/2}} $
C) $ {\log_{e}}{{( \frac{x}{2-x} )}^{1/2}} $
D) $ {\log_{e}}{{( \frac{x-1}{x+1} )}^{-2}} $
Show Answer
Answer:
Correct Answer: B
Solution:
$ y=\frac{e^{x}-{e^{-x}}}{e^{x}+{e^{-x}}}+2\Rightarrow y=\frac{e^{2x}-1}{e^{2x}+1}+2 $
$ \Rightarrow e^{2x}=\frac{1-y}{y-3}=\frac{y-1}{3-y}\Rightarrow x=\frac{1}{2}{\log_{e}},( \frac{y-1}{3-y} ) $
$ \Rightarrow {f^{-1}}(y)={\log_{e}},{{( \frac{y-1}{3-y} )}^{1/2}}\Rightarrow {f^{-1}}(x)={\log_{e}}{{( \frac{x-1}{3-x} )}^{1/2}} $ .
BETA
