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

Dec 3, 2017

Domain: $\text{ } x < - 1 \mathmr{and} x > - 1$

Range: $\text{ } f \left(x\right) < 0 \mathmr{and} f \left(x\right) > 0$

Using interval notations

Domain:$\text{ } \left(- \infty , - 1\right) \cup \left(- 1 , \infty\right)$

Range:$\text{ } \left(- \infty , 0\right) \cup \left(0 , \infty\right)$

#### Explanation:

We are given the function $f \left(x\right) = - \left(\frac{1}{x + 1}\right)$

To find the Domain of $f \left(x\right)$, set $\textcolor{g r e e n}{x + 1 = 0}$

$\Rightarrow \left(x + 1\right) = 0$

$\Rightarrow x = - 1$

We must remember that $x = - 1$ is our Vertical Asymptote

So, our Domain is $\left(- \infty , - 1\right) \cup \left(- 1 , \infty\right)$

To find the Range of $f \left(x\right)$

Let us now observe our original function $f \left(x\right) = - \left(\frac{1}{x + 1}\right)$

We note that there is no x term in the numerator(NR)

Hence, the degree of numerator is ZERO(0).

The highest degree of the denominator(DR) is ONE(1).

We observe that degree of the DR $>$ degree of the NR

Hence, we conclude that $y = 0$ is our Horizontal Asymptote

Range: $\text{ } f \left(x\right) < 0 \mathmr{and} f \left(x\right) > 0$

Range using interval notations is $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$

Please investigate the graph for a visual comprehension