# How do you state the domain and range of f(x)=x/(x-1)?

Nov 30, 2016

The domain is x in ] -oo,1 [ uu ] 1,+oo[

The range is f(x) in ] -oo,1 [ uu ] 1,+oo [

#### Explanation:

As you cannot divide by $0$, $x \ne 1$

So the domain is x in ] -oo,1 [ uu ] 1,+oo[

For the limits $x \to \pm \infty$, we take the terms of highest degree in the numerator and the deniminator

${\lim}_{x \to \pm \infty} f \left(x\right) = {\lim}_{x \to \pm \infty} \frac{x}{x} = 1$

${\lim}_{x \to {1}^{-}} f \left(x\right) = \frac{1}{0} ^ \left(-\right) = - \infty$

${\lim}_{x \to {1}^{+}} f \left(x\right) = \frac{1}{0} ^ \left(+\right) = + \infty$

So the range is f(x) in ] -oo,1 [ uu ] 1,+oo [

graph{x/(x-1) [-10, 10, -5, 4.995]}