Given f(x)=rootsignx-2 and g(x) = x-7 what is the domain of the quotient function?
A. (-infinity,2]
B. )-infinity,7 U (7, infinity)
C. [2,7) U (7, infinity)
D. (2,infinity)
To find the domain of the quotient function, we need to consider the domains of the individual functions and any restrictions imposed by the operations.
The domain of a square root function (f(x) = √(x - 2)) is all the real numbers that make the radicand (the expression under the square root) non-negative. In this case, x - 2 must be greater than or equal to 0, so x ≥ 2. Therefore, the domain of f(x) is [2, infinity).
The domain of a linear function (g(x) = x - 7) is all real numbers, as there are no restrictions.
For the quotient function (h(x) = f(x) / g(x)), we must also consider any potential division by zero. Since the denominator g(x) = x - 7 is a linear function, it will not be zero for any real numbers. Therefore, there are no additional restrictions on the domain due to division by zero.
Combining the domains of f(x) and g(x), and considering the absence of division by zero restrictions, we have:
Domain of h(x) = Domain of f(x) ∩ Domain of g(x) = [2, infinity) (intersection) (-infinity, infinity) = [2, infinity).
Therefore, the correct answer is (D) (2, infinity).