if-f-x-10-x-10-x-10-x-10-x-then-f-1-x-

Question Number 133893 by mathlove last updated on 25/Feb/21

i f f (x) = \frac{10^{x} - 10^{- x}}{10^{x} + 10^{- x}} t h e n f^{- 1} (x) = ?

Answered by rs4089 last updated on 25/Feb/21

y = \frac{10^{x} - 10^{- x}}{10^{x} + 10^{- x}} \Rightarrow \frac{1 + y}{1 - y} = \frac{10^{x}}{10^{- x}}

10^{2 x} = \frac{1 + y}{1 - y}

x = \frac{1}{2} {l o g}_{10} (\frac{1 + y}{1 - y})

x \to f^{- 1} (x) a n d y \to x

f^{- 1} (x) = \frac{1}{2} {l o g}_{10} (\frac{1 + x}{1 - x})

Answered by TheSupreme last updated on 25/Feb/21

10^{x} = e^{l n (10) x}

f (x) = \frac{\frac{e^{l n (10) x} - e^{l n (10) x}}{2}}{\frac{e^{l n (10) x} + e^{l n (10) x}}{2}} = \frac{s i n h (l n (10) x)}{c o s h (l n (10) x)} = t a n h (l n (10) x)

l n (10) x = {t a n h}^{- 1} (y)

x = \frac{{t a n h}^{- 1} (y)}{l n (10)}