# How do you find the domain of f(x)=(2x+1)/(x-3)?

May 18, 2018

$x \in \left(- \infty , 3\right) \cup \left(3 , + \infty\right)$

#### Explanation:

The domain is the set of $x$ values that are defined for the function $f \left(x\right)$.

Naturally a continuous function will have a domain $x \in \left(- \infty , + \infty\right)$. However, some functions have discontinuities. These are values of $x$ for which the function is not properly defined.

With rational functions, these invalid $x$ values occur when the denominator is $0$, as division by zero is undefined.

We have $x - 3 = 0 \to x = 3$ as a value of $x$ that our function cannot take. We write this new domain as a combination of domains extending to infinity and containing all $x$ values as close to $3$ as possible without including $3$ itself:

$x \in \left(- \infty , 3\right) \cup \left(3 , + \infty\right)$