# What is the domain and range of (x+5)/(x+1)?

May 25, 2016

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

#### Explanation:

First of all, we must note that this is a reciprocal funtion, as it has $x$ in the lower part of the division. Therefore, it will have a domain restiction:

$x + 1 \ne 0$
$x \ne 0$

The division by zero is not defined in mathematics, so this function will not hava a value associated to $x = - 1$. There will be two curves that pass near this point, so we can procced to plot this function for points around this restriction:

$f \left(- 4\right) = \frac{1}{-} 3 = - 0.333$
$f \left(- 3\right) = \frac{2}{-} 2 = - 1$
$f \left(- 2\right) = \frac{3}{-} 1 = - 3$
$f \left(- 1\right) = \cancel{\exists}$
$f \left(0\right) = \frac{5}{1} = 5$
$f \left(1\right) = \frac{6}{2} = 3$
$f \left(2\right) = \frac{7}{3} = 2.333$

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

There is also a hidden range restriction in this function. Notice that the curves will keep going towards the infinitity in both sides by the x axis, but they never reach a value. We must calculate the limits of the function in both infinities:

${\lim}_{x \to + \infty} f = 1$

${\lim}_{x \to - \infty} f = 1$

This number can be found if you solve the function for a very big number in x (1 million, for example) and a very small number (-1 million). The funcion will get near $y = 1$, but the result will never be exactly 1.

Finally, the domain can be any number, except -1, so we write it this way: RR-{-1.
The range can be any number except 1: #RR-{1}.