if-f-x-ln-x-x-2-1-find-f-1-x-

Question Number 66412 by aliesam last updated on 14/Aug/19

i f

f (x) = l n (x + \sqrt{x^{2} + 1})

f i n d

f^{- 1} (x) = ?

Commented by mathmax by abdo last updated on 14/Aug/19

w e h a v e f (s h (x)) = l n (s h (x) + \sqrt{1 + {s h}^{2} x}) = l n (s h (x) + c h (x))

= l n (\frac{e^{x} - e^{- x}}{2} + \frac{e^{x} + e^{- x}}{2}) = l n (e^{x}) = x \Rightarrow f^{- 1} (x) = s h (x)

Commented by mr W last updated on 14/Aug/19

f (x) = l n (x + \sqrt{x^{2} + 1}) = \sinh^{- 1} x

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

Commented by kaivan.ahmadi last updated on 15/Aug/19

y = l n (x + \sqrt{x^{2} + 1}) \Rightarrow e^{y} - x = \sqrt{x^{2} + 1} \Rightarrow e^{2 y} - 2 {x e}^{y} + x^{2} = x^{2} + 1 \Rightarrow

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

Answered by MJS last updated on 14/Aug/19

x = \ln (y + \sqrt{y^{2} + 1})

e^{x} = y + \sqrt{y^{2} + 1}

{(e^{x} - y)}^{2} = y^{2} + 1

e^{2 x} - 2 y e^{x} + y^{2} = y^{2} + 1

y = \frac{e^{x} - e^{- x}}{2} = \sinh x

Commented by Mikael last updated on 15/Aug/19

n i c e S i r .

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