# What is the domain and range of the given function f(x)= (x-1)/(x+3)?

Feb 18, 2017

Domain: $\left(- \infty , - 3\right) U \left(- 3 , \infty\right)$
Range: $\left(- \infty , 1\right) U \left(1 , \infty\right)$

#### Explanation:

Rational function: $\frac{N \left(x\right)}{D \left(x\right)} = \frac{x - 1}{x + 3}$:

Analytically, vertical asymptotes are found when you set $D \left(x\right) = 0$:

$x + 3 = 0$; $x = - 3$ so the vertical asymptote is at $x = - 3$

Horizontal asymptotes are found based on the degree of the functions: $\frac{a {x}^{n}}{b {x}^{m}}$ When $n = m , y = \frac{a}{b} = 1$

so the horizontal asymptote is at $y = 1$

You can see this from the graph:
graph{(x-1)/(x+3) [-10, 10, -5, 5]}