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1.Relation and Function
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The inverse of the function $\frac{{{{10}^x} - {{10}^{ - x}}}}{{{{10}^x} + {{10}^{ - x}}}}$ is
A
$\frac{1}{2}{\log _{10}}\left( {\frac{{1 + x}}{{1 - x}}} \right)$
B
$\frac{1}{2}{\log _{10}}\left( {\frac{{1 - x}}{{1 + x}}} \right)$
C
$\frac{1}{4}{\log _{10}}\left( {\frac{{2x}}{{2 - x}}} \right)$
D
None of these
Solution
(a) $y = \frac{{{{10}^x} – {{10}^{ – x}}}}{{{{10}^x} + {{10}^{ – x}}}}$
$\Rightarrow x = \frac{1}{2}{\log _{10}}\left( {\frac{{1 + y}}{{1 – y}}} \right)$
Let $y = f(x)$ ==> $x = \pi ,\,\,f(\pi ) = – \tan \frac{\pi }{4} = – 1$
==> ${f^{ – 1}}(y) = \frac{1}{2}{\log _{10}}\left( {\frac{{1 + y}}{{1 – y}}} \right)$
==>${f^{ – 1}}(x) = \frac{1}{2}{\log _{10}}\left( {\frac{{1 + x}}{{1 – x}}} \right)$
Standard 12
Mathematics
