# How do you find the domain and range of y = (x+4 )/( x-4)?

Oct 16, 2015

Domain: $\left(- \infty , 4\right) \cup \left(4 , + \infty\right)$
Range: $\left(- \infty , 1\right) \cup \left(1 , + \infty\right)$

#### Explanation:

The domain of the function will include all the values of $x$ for which the denominator is not equal to zero.

This means that you have

$x - 4 = 0 \implies x = 4$

This value of $x$ will be excluded from the domain of the function. This implies that the function's domain will be $x \in \mathbb{R} \text{\} \left\{4\right\}$, or $x \in \left(- \infty , 4\right) \cup \left(4 , + \infty\right)$.

To find the range of the function, use some algebraic manipulation to rewrite the function as

$y = \frac{x + 4}{x - 4} = \frac{x - 4 + 8}{x - 4} = 1 + \frac{8}{x - 4}$

Since $\frac{8}{x - 4} \ne 0 \forall x \in \left(- \infty , 4\right) \cup \left(4 , + \infty\right)$, it follows that you can never have $y = 1$.

This means that the range of the function will be $x \in \mathbb{R} \text{\} \left\{1\right\}$, or $x \in \left(- \infty , 1\right) \cup \left(1 , + \infty\right)$.

graph{(x+4)/(x-4) [-9.99, 10.02, -5, 4.995]}