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1.Relation and Function
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જો વિધેય $f : R \to R$ માટે $f(x) = log_a(x + \sqrt {x^2 +1} ), (a > 0, a \neq 1)$ હોય તો $f^{-1}(x)$ =
A
$\left( {\frac{{{a^x} + {a^{ - x}}}}{2}} \right)$
B
$\left( {\frac{{{a^x} - {a^{ - x}}}}{2}} \right)$
C
$\forall x \in R$ માટે શક્ય નથી
D
માત્ર $x \in R^+$ માટે શક્ય છે
Solution
$f(x)=\log _{a}\left(x+\sqrt{x^{2}+1}\right)$
let $f(x)=y$
$x=F^{-1}(y)$
$y=\log _{a}\left(x+\sqrt{x^{2}+1}\right)$
$2+\sqrt{x^{2}+1}=a^{y}$
$\left(x-a^{y}\right)^{2}=x^{2}+1$
$x^{2}+a^{2 y}-2 x a^{y}=x^{2}+1$
$a^{2 y}-1=2 x a^{y}$
$x=\frac{a^{2 y}-1}{2 a y}$
$f^{-1}(y)=\frac{a^{2 y}-1}{2 a^{y}}$
$f^{-1}(x)=\frac{a^{2 x}-1}{2 a^{x}}$
Standard 12
Mathematics
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