First slide

Functions (XII)

Question

The inverse of the function $f (x) = \frac{a^{x} - a^{- x}}{a^{x} + a^{- x}} is$

Moderate

A

$\frac{1}{2} \log_{a} (\frac{1 - x}{1 + x})$
B

$\frac{1}{2} \log_{a} (\frac{1 + x}{1 - x})$
C

$\log_{a} (\frac{1 + x}{1 - x})$
D

None of these

Solution

$Let y = \frac{a^{x} - a^{- x}}{a^{x} + a^{- x}} = \frac{a^{2 x} - 1}{a^{2 x} + 1} \Rightarrow \frac{y - 1}{y + 1} = \frac{(a^{2 x} - 1) - (a^{2 x} + 1)}{(a^{2 x} - 1) + (a^{2 x} + 1)}$
(Using componendo and dividendo)
$\begin{array}{l} \Rightarrow \frac{y - 1}{y + 1} = \frac{- 2}{2 a^{2 x}} \Rightarrow a^{2 x} = \frac{1 + y}{1 - y} \\ \Rightarrow 2 {xlog}_{a} a = \log_{a} (\frac{1 + y}{1 - y}) \\ \Rightarrow x = \frac{1}{2} \log_{a} (\frac{1 + y}{1 - y}) \end{array}$

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