If g(x) = 3x - 1 and f(x) = 9−𝑥29−x 2 , then which of the following will have a domain of [−3,1/3)∪(1/3,3][−3,1/3)∪(1/3,3]?A.(f/g)(x)B.g(f(x))C.(g - f)(x)D.(g + f)(x)E.f(g(x))SUBMITarrow_backPREVIOUS
Question
If g(x) = 3x - 1 and f(x) = 9−𝑥29−x 2 , then which of the following will have a domain of [−3,1/3)∪(1/3,3][−3,1/3)∪(1/3,3]?A.(f/g)(x)B.g(f(x))C.(g - f)(x)D.(g + f)(x)E.f(g(x))SUBMITarrow_backPREVIOUS
Solution
The domain of a function is the set of all possible input values (often referred to as 'x' values) which will output real numbers.
Let's analyze each option:
A. (f/g)(x) = f(x) / g(x) = (9−x^2) / (3x - 1) The denominator cannot be zero, so x ≠ 1/3. The numerator is a quadratic function, which is defined for all real numbers. Therefore, the domain is all real numbers except 1/3.
B. g(f(x)) = g(9 - x^2) = 3(9 - x^2) - 1 This is a composition of two functions, both of which are defined for all real numbers. Therefore, the domain is all real numbers.
C. (g - f)(x) = g(x) - f(x) = (3x - 1) - (9 - x^2) Both functions are defined for all real numbers, so the domain is all real numbers.
D. (g + f)(x) = g(x) + f(x) = (3x - 1) + (9 - x^2) Both functions are defined for all real numbers, so the domain is all real numbers.
E. f(g(x)) = f(3x - 1) = 9 - (3x - 1)^2 This is a composition of two functions. The inner function g(x) = 3x - 1 is defined for all real numbers. The outer function f(x) = 9 - x^2 is also defined for all real numbers. However, the composition f(g(x)) is not defined when (3x - 1)^2 > 9, which gives us the intervals x < -3 or 1/3 < x < 3.
Therefore, the correct answer is E. f(g(x)).
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