Mathematik-Online lexicon: Example: Inverse Function

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	Mathematik-Online lexicon:
	Example: Inverse Function

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Example:

Determine the inverse function $f^{-1}(x)$ of the function $f(x)=1-\frac{2e^x}{e^x+1}$ , with $D_f=\mathbb{R}$ and $\operatorname{Range}(f)=\mathbb{R}$ .

Domain and range of $f^{-1}(x)$ can be determined at once:

$D_{f^{-1}}=\mathbb{R}$ and $\operatorname{Range}(f^{-1})=\mathbb{R}$

Write instead of :

$\displaystyle y=1-\frac{2e^x}{e^x+1}$

$\displaystyle 1-y$	$\displaystyle =\frac{2e^x}{e^x+1}$
$\displaystyle 2e^x$	$\displaystyle =e^x+1-ye^x-y$
$\displaystyle e^x+ye^x$	$\displaystyle =1-y$
$\displaystyle e^x(1+y)$	$\displaystyle =1-y$
$\displaystyle e^x$	$\displaystyle =\frac{1-y}{1+y}$
$\displaystyle x$	$\displaystyle =\ln\left(\frac{1-y}{1+y}\right)$