# What is the domain and range of (x-1)/(x-4)?

Aug 29, 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 possible value of $x$ except the value that makes the denominator equal to zero. More specifically, $x = 4$ will be excluded from the domain, which will thus be $\left(- \infty , 4\right) \cup \left(4 , + \infty\right)$.

To determine the range of the function, you can do a little algebraic manipulation to rewrite the function as

$y = \frac{\left(x - 4\right) + 3}{x - 4} = 1 + \frac{3}{x - 4}$

Since the fraction $\frac{3}{x - 4}$ can never be equal to zero, the function can never take the value

$y = 1 + 0 = 1$

This means that the range of the function will be $\left(- \infty , 1\right) \cup \left(1 , + \infty\right)$.

graph{(x-1)/(x-4) [-18.8, 21.75, -10.3, 9.98]}