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.g(f(x))B.(g + f)(x)C.(f/g)(x)D.f(g(x))E.(g - f)(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.g(f(x))B.(g + f)(x)C.(f/g)(x)D.f(g(x))E.(g - f)(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. g(f(x)): The domain is all real numbers because the function f(x) = 9−𝑥2 is defined for all real numbers and g(x) = 3x - 1 is also defined for all real numbers.
B. (g + f)(x): The domain is all real numbers because both g(x) and f(x) are defined for all real numbers.
C. (f/g)(x): The domain is all real numbers except those for which g(x) = 0, because we cannot divide by zero. Solving the equation 3x - 1 = 0, we find that x = 1/3. So, the domain of (f/g)(x) is all real numbers except x = 1/3.
D. f(g(x)): The domain is all real numbers because g(x) is defined for all real numbers and the output of g(x) is the input for f(x), which is also defined for all real numbers.
E. (g - f)(x): The domain is all real numbers because both g(x) and f(x) are defined for all real numbers.
So, the correct answer is C. (f/g)(x), because its domain is all real numbers except x = 1/3, which matches the given domain [−3,1/3)∪(1/3,3].
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