# How do you find domain and range for  f(x)= x/(x-2)?

Oct 1, 2015

Domain: $\mathbb{R} - \left\{2\right\}$
Range: $\mathbb{R} - \left\{1\right\}$

#### Explanation:

$f \left(x\right) = \frac{2}{x} - 2$ is clearly defined for all Real values of $x$ except $x = 2$ (since division by $0$ is undefined).
Therefore the Domain is all Real values except $2$

To determine the range, consider the possible limitations on $g \left(x\right)$ where $g \left(x\right)$ is the inverse of $f \left(x\right)$

By definition of inverse
$\textcolor{w h i t e}{\text{XXX}} f \left(g \left(x\right)\right) = x$
and from the given definition of $f \left(x\right)$
$\textcolor{w h i t e}{\text{XXX}} f \left(g \left(x\right)\right) = g \frac{x}{g \left(x\right) - 2}$

Therefore
$\textcolor{w h i t e}{\text{XXX}} g \frac{x}{g \left(x\right) - 2} = x$

$\textcolor{w h i t e}{\text{XXX}} g \left(x\right) = x \cdot g \left(x\right) - x \cdot 2$

$\textcolor{w h i t e}{\text{XXX}} g \left(x\right) - x \cdot g \left(x\right) = - 2 x$

$\textcolor{w h i t e}{\text{XXX}} g \left(x\right) \left(1 - x\right) = - 2 x$

$\textcolor{w h i t e}{\text{XXX}} g \left(x\right) = \frac{- 2 x}{1 - x}$

which is defined for all values of $x \ne 1$

Therefore the Range is all Real values except $1$
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