# What is the domain and range of y = (x + 3) / (x -5)?

Mar 17, 2018

Domain: $\left(- \infty , 5\right) \cup \left(5 , \infty\right)$
Range: $\left(- \infty , 1\right) \cup \left(1 , \infty\right)$

#### Explanation:

The domain of this equation is all numbers except when you divide by $0$. So we need to find out at what $x$ values does the denominator is equal to $0$. To do this we simply we the denominator equal to $0$. Which is

$x - 5 = 0$

Now we get $x$ alone by adding $5$ is both sides, giving us
$x = 5$

So at $x = 5$ this function is undefined.
That means that every other number you can think of will be valid for this function. Which gives us $\left(- \infty , 5\right) \cup \left(5 , \infty\right)$

Now to find the Range
The range can be found by dividing the leading coefficients from the numerator and the denominator. In the numerator we have $x + 3$ and in the denominator we have $x - 5$

Since there is no number in front of the $x$ values we just treat it as $1$

So it would $\frac{1}{1}$ which is $1$.
So the range is $\left(- \infty , 1\right) \cup \left(1 , \infty\right)$