Question

How do you find the domain of `g(x)=6/(9-5x)`?

Answer 1

`x inRR,x!=9/5`

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 "9-5x=0rArrx=9/5larrcolor(red)"excluded value"`

`"domain is "x inRR,x!=9/5`

`(-oo,9/5)uu(9/5,oo)larrcolor(blue)"in interval notation"`
graph{6/(9-5x) [-10, 10, -5, 5]}

Answer 2

`x inRR, x!=9/5`

Explanation:

The only thing that will make `g(x)` undefined is when the denominator is zero, so let's set it to zero to find any excluded values in the domain.

`-5x+9=0=>-5x=-9=>x=9/5`

The value `x=9/5` is not included in our domain, so we can say

`x inRR, x!=9/5`

We can even see this graphically, as we have a vertical asymptote at `x=9/5`.

graph{6/(9-5x) [-10, 10, -5, 5]}

Hope this helps!