Question

f(x)=√x+3 and g(x)= 1/(x-4). Simplify and state the domain of f/g

Answer 1

The domain is all the values of x that the function can take on, right?
since f/g means f(x)/g(x) you need to test for g(x) = 0 in order to determine which values would cause the function to become undefined.
SO! Take \[\frac{ 1 }{ x-4 }=0\] and solve for x giving you x = 4. This means that x can NOT be 4, otherwise f/g becomes undefined.
Note that your numerator has a square root in it so x has to be positive.
So your domain has to be positive and not 4.
Gives you: \[x \epsilon \mathbb{R} : 0 \le x < 4 \cup x > 4\]

Answer 2

Could also say:
Domain = [0,4) U (4,inf)