f(x)=root sign, and inside that x-2 and g(x)x-7. Which of the following 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)
sqrt(x-2) /( x-7) ???
x must be 2 or greater or we have the square root of a negative number, which is not real.
x may not be 7 or we have a zero in the denominator
therefore
2 to 7 and 7 to + infinity
C
Is C the quotient function of f/g?
yes
To determine the domain of the quotient function, we need to consider the restrictions on the division operation and the square root function.
Let's first consider the restrictions on the division operation. In general, we cannot divide by zero, so the denominator of the quotient function must be non-zero.
In this case, the denominator is g(x) = x - 7. Thus, we need to find the values of x for which x - 7 ≠ 0.
Solving this inequality, we get:
x ≠ 7
Therefore, the function is defined for all values of x except x = 7. So, the domain of the quotient function is (-∞, 7) U (7, ∞).
Now, let's consider the restrictions on the square root function. The argument inside the square root, f(x) = √(x - 2), must be greater than or equal to zero.
Solving this inequality, we get:
x - 2 ≥ 0
x ≥ 2
Therefore, the function is defined for all values of x greater than or equal to 2. So, the domain of the quotient function is [2, ∞).
Now, let's find the common domain by considering both restrictions. The domain of the quotient function is determined by the intersection of the individual domains. In this case, the intersection of (-∞, 7) U (7, ∞) and [2, ∞) is (2, ∞).
Therefore, the correct answer is D. (2, ∞).