# What is the range of y=(-4x-3)/(x-2)?

Jun 4, 2015

$f \left(x\right) = y = \frac{- 4 x - 3}{x - 2}$

$= \frac{\left(- 4 x + 8\right) - 11}{x - 2}$

$= \frac{- 4 \left(x - 2\right) - 11}{x - 2}$

$= - 4 - \frac{11}{x - 2}$

$f \left(x\right)$ can take any value except $- 4$ (to which $f \left(x\right)$ is asymptotic as $x \to \pm \infty$).

So the range is $\mathbb{R}$ \ $\left\{- 4\right\}$

or in interval notation: $\left(- \infty , - 4\right) \cup \left(- 4 , \infty\right)$

graph{(-4x-3)/(x-2) [-39.76, 39.8, -19.92, 19.83]}

More explicitly, we can express $x$ in terms of $y$ as

$x = 2 - \frac{11}{y + 4}$

which has an obvious excluded value at $y = 4$