The domain of h(x)=f(x)/g(x) is x≠2, x≠1.

Question

Which of the following are possible? Select all that apply.

g contains the factor (x−2)

f contains the factor (x−2)

g(2)=0

g(−1)=0

Answer 1

Let's go through all of them one-by-one:

g contains the factor (x-2). If g contains the factor (x-2), then if x = 2, g(2) = 2-2 = 0. Since we cannot divide by 0, this function value is undefined, so x = 2 cannot be in the domain. So this answer is right.

f contains the factor (x-2). If f contains this factor, we just get 0 in the numerator, so we get that h(2) = 0/something = 0. Not undefined, so no domain issue. So this is wrong.

g(2) = 0. If g(2) = 0, then if we plug it into h, we are dividing by zero, so the function value is undefined at that point, so x = 2 cannot be in the domain. So this answer is right.

g(-1) = 0. Unless you made a typo, the domain is x ≠ 1 or x ≠ 2, since we are talking about x = -1, this is of no relevance to our domain. So this answer is wrong.

a) and c) are the right answers.