f-x-ln-x-x-2-1-find-f-1-x-plzz-help-

Question Number 27608 by chernoaguero@gmail.com last updated on 10/Jan/18

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

find f^{- 1} (x)

plzz help

Commented by abdo imad last updated on 11/Jan/18

f (x) = y \Leftrightarrow f^{- 1} (y) = x b u t f (x) = y \Leftrightarrow l n (x + \sqrt{x^{2} + 1}) = y

x + \sqrt{x^{2} + 1} = e^{y} \Leftrightarrow \sqrt{x^{2} + 1} = e^{y} - x a n d e^{y} > x

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

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

\Leftrightarrow x = \frac{e^{2 y} - 1}{2 e^{y}} = \frac{e^{y} - e^{- y}}{2} = s i n h y

f^{- 1} (x) = \frac{e^{x} - e^{- x}}{2} = s h x .

Commented by chernoaguero@gmail.com last updated on 11/Jan/18

Thank u sir but what do u mean

by sinhy an shx

Commented by Joel578 last updated on 11/Jan/18

sh x is \sinh x

Commented by abdo imad last updated on 11/Jan/18

s i n h x = s h x i t s o n l y a n o t a t i o n .

Commented by chernoaguero@gmail.com last updated on 11/Jan/18

ok thank