If-f-x-10-x-10-x-10-x-10-x-find-f-1-x-

Question Number 143595 by ZiYangLee last updated on 16/Jun/21

If f (x) = \frac{10^{x} - 10^{- x}}{10^{x} + 10^{- x}}, find f^{- 1} (x) .

Answered by bobhans last updated on 16/Jun/21

l e t 10^{x} = t

\Rightarrow y = f (t) = \frac{t - \frac{1}{t}}{t + \frac{1}{t}} = \frac{t^{2} - 1}{t^{2} + 1}

\Rightarrow (y - 1) t^{2} = - y - 1

\Rightarrow t = \pm \sqrt{\frac{- y - 1}{y - 1}} = \pm \sqrt{\frac{y + 1}{1 - y}}

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

\Rightarrow x = \log_{10} (\sqrt{\frac{y + 1}{1 - y}})

\Rightarrow f^{- 1} (x) = \log_{10} (\sqrt{\frac{x + 1}{1 - x}})

Commented by ZiYangLee last updated on 16/Jun/21

thank you very much!

Answered by qaz last updated on 16/Jun/21

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

\Rightarrow x = \frac{1}{2} \lg \frac{1 + y}{1 - y}

\Rightarrow f^{- 1} (x) = \frac{1}{2} \lg \frac{1 + x}{1 - x} \dots . . (∣ x ∣< 1)

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