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

Jun 2, 2017

The range of $\frac{x - 1}{x - 4}$ is $\mathbb{R} \text{\} \left\{1\right\}$ a.k.a. $\left(- \infty , 1\right) \cup \left(1 , \infty\right)$

#### Explanation:

Let:

$y = \frac{x - 1}{x - 4} = \frac{x - 4 + 3}{x - 4} = 1 + \frac{3}{x - 4}$

Then:

$y - 1 = \frac{3}{x - 4}$

Hence:

$x - 4 = \frac{3}{y - 1}$

Adding $4$ to both sides, we get:

$x = 4 + \frac{3}{y - 1}$

All these steps are reversible, except division by $\left(y - 1\right)$, which is reversible unless $y = 1$.

So given any value of $y$ apart from $1$, there is a value of $x$ such that:

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

That is, the range of $\frac{x - 1}{x - 4}$ is $\mathbb{R} \text{\} \left\{1\right\}$ a.k.a. $\left(- \infty , 1\right) \cup \left(1 , \infty\right)$

Here's the graph of our function with its horizontal asymptote $y = 1$

graph{(y-(x-1)/(x-4))(y-1) = 0 [-5.67, 14.33, -4.64, 5.36]}

If the graphing tool allowed, I would also plot the vertical asymptote $x = 4$

Jun 2, 2017

$y \in \mathbb{R} , y \ne 1$

#### Explanation:

$\text{rearrange "y=(x-1)/(x-4)" making x the subject}$

$\Rightarrow y \left(x - 4\right) = x - 1 \leftarrow \textcolor{b l u e}{\text{ cross-multiplying}}$

$\Rightarrow x y - 4 y = x - 1$

$\Rightarrow x y - x = - 1 + 4 y$

$\Rightarrow x \left(y - 1\right) = 4 y - 1$

$\Rightarrow x = \frac{4 y - 1}{y - 1}$

$\text{the denominator of x cannot be zero as this would make}$
$\text{x undefined.}$

$\text{equating the denominator to zero and solving gives the}$
$\text{value that y cannot be}$

$\text{solve " y-1=0rArry=1larrcolor(red)" excluded value}$

$\Rightarrow \text{range is } y \in \mathbb{R} , y \ne 1$