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

Aug 16, 2015

Domain: $\left(- \infty , - 2\right) \cup \left(- 2 , + \infty\right)$
Range: $\left(- \infty , 0\right) \cup \left(0 , + \infty\right)$

#### Explanation:

First, notice that you can rewrite your function as

$f \left(x\right) = \frac{1}{2 \cdot \left(x + 2\right)}$

This function is defined for any value of $x \in \mathbb{R}$ except the value that would make the denominator equal to zero.

More specifically, you need to exclude from the domain of the function the value of $x$ that would make

$x + 2 = 0 \implies x = - 2$

Therefore, the domain of the function will be $\mathbb{R} - \left\{- 2\right\}$, or $\left(- \infty , - 2\right) \cup \left(- 2 , + \infty\right)$.

Notice that since you're dealing with a fraction that has a constant numerator, the function has no way of ever being equal to zero.

$f \left(x\right) \ne 0 \text{, } \left(\forall\right) x \in \mathbb{R} - \left\{- 2\right\}$

The range of the function will thus be $\mathbb{R} - \left\{0\right\}$, or $\left(- \infty , 0\right) \cup \left(0 , + \infty\right)$.

graph{1/(2x + 4) [-6.243, 6.243, -3.12, 3.123]}