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1.Relation and Function
normal
Let f : $R \to R$ be defined by $f\left( x \right) = \ln \left( {x + \sqrt {{x^2} + 1} } \right)$ , then number of solutions of $\left| {{f^{ - 1}}\left( x \right)} \right| = {e^{ - \left| x \right|}}$ is
A
$1$
B
$2$
C
$3$
D
Infinite
Solution
$f(x)=y=\ln (x+\sqrt{x^{2}+1})$
$\Rightarrow \mathrm{e}^{y}=\sqrt{\mathrm{x}^{2}+1}+\mathrm{x} ; \mathrm{e}^{-\mathrm{y}}=\sqrt{\mathrm{x}^{2}+1}-\mathrm{x}$
$f^{-1}(x)=\frac{e^{x}-e^{-x}}{2}$
No. of solutions $=2$
Standard 12
Mathematics
