# How do you find the domain and range of f(x) = (x-4)/(x+3) ?

Mar 7, 2017

The domain is $x \ne - 3$ as this would make the denominator $= 0$.
There are no other restrictions.

#### Explanation:

For the range we look at what happens when $x \to \pm \infty$
In both cases the fraction will approach $f \left(x\right) = + 1$, but will never get there. $y \ne + 1$

In short:
${\lim}_{x \to - {3}^{-}} f \left(x\right) = + \infty$ and ${\lim}_{x \to - {3}^{+}} f \left(x\right) = - \infty$

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

$x = - 3$ and $y = + 1$ are called the asymptotes .
graph{(x-4)/(x+3) [-25.65, 25.65, -12.82, 12.84]}