# What is the domain and range of f(x)= (x+1)/(x^2+3x-4)?

Sep 21, 2015

Domain: $\mathbb{R} - \left\{4 , + 1\right\}$
Range: $\mathbb{R}$

#### Explanation:

Given $f \left(x\right) = \frac{x + 1}{{x}^{2} + 3 x - 4}$

Notice that the denominator can be factored as
$\textcolor{w h i t e}{\text{XXX}} \left(x + 4\right) \left(x - 1\right)$
which implies that the denominator would be $0$ if $x = - 4$ or $x = 1$
and since division by $0$ is undefined
the Domain must exclude these values.

For the Range:
Consider the graph of $f \left(x\right)$
graph{(x+1)/(x^2+3x-4) [-10, 10, -5, 5]}
It seems clear that all values of $f \left(x\right)$ (even within $x \in \left(- 4 , + 1\right)$) can be generated by this relation.
Therefore the Range of $f \left(x\right)$ is all Real numbers, $\mathbb{R}$