# Given the function g(x)=(x^2-3x-4)/(x-5), how do you find the domain?

Jul 29, 2018

$x \in \mathbb{R} , x \ne 5$

#### Explanation:

The denominator of g(x) cannot be zero as this would make g(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be.

$\text{solve "x-5=0rArrx=5larrcolor(red)"excluded value}$

$\text{domain is } x \in \mathbb{R} , x \ne 5$

$\left(- \infty , 5\right) \cup \left(5 , \infty\right) \leftarrow \textcolor{b l u e}{\text{in interval notation}}$
graph{(x^2-3x-4)/(x-5 [-40, 40, -20, 20]}

Jul 30, 2018

$x \in \mathbb{R} , x \ne 5$

#### Explanation:

The only $x$ value that will make this function undefined is when the denominator is set to zero. We see that this value is $x = 5$.

Therefore, we can say that the domain is $x \in \mathbb{R} , x \ne 5$. This is just a fancy way of saying $x$ can be any real number except $5$.

We also see this graphically, as we have a vertical asymptote at $x = 5$.

graph{(x^2-3x-4)/(x-5) [-74, 86, -36.8, 43.2]}

Hope this helps!