Question

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

Answer 1

`x inRR,x!=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.

`"solve "x-5=0rArrx=5larrcolor(red)"excluded value"`

`"domain is "x inRR,x!=5`

`(-oo,5)uu(5,oo)larrcolor(blue)"in interval notation"`
graph{(x^2-3x-4)/(x-5 [-40, 40, -20, 20]}

Answer 2

`x inRR, x!=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 inRR, x!=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!