The domain of f/g consists of numbers x for which g(x) cannot equal 0 that are in the domains of both f and g.
Let’s take this equation as an example:
If f(x) = 3x - 5 and g(x) = square root of x-5, what is the domain of (f/g)x.
For x to be in the domain of (f/g)(x), it must be in the domain of f and in the domain of g since (f/g)(x) = f(x)/g(x). We also need to ensure that g(x) is not zero since f(x) is divided by g(x). Therefore, there are 3 conditions.
x must be in the domain of f: f(x) = 3x -5 are in the domain of x and all real numbers x.
x must be in the domain of g: g(x) = √(x - 5) so x - 5 ≥ 0 so x ≥ 5.
g(x) can not be 0: g(x) = √(x - 5) and √(x - 5) = 0 gives x = 5 so x ≠ 5.
Hence to x x ≥ 5 and x ≠ 5 so the domain of (f/g)(x) is all x satisfying x > 5.
Thus, satisfying satisfy all three conditions, x x ≥ 5 and x ≠ 5 so the domain of (f/g)(x) is all x satisfying x > 5.
