Final answer:
The correct answer to the question "Given f(x)=3x and g(x)= 1/x+3 , which value is in the domain of f∗g? A) x=−3 B) x=0 C) x=1 D) x=−1" is actually (D) x=-1.
Explanation:
Composition Domain: The domain of f ∘ g (composition of f and g) is the set of all values in the domain of g that make g(x) fall within the domain of f.
g(x) Domain Restriction: g(x) = 1/x + 3 is undefined when x = 0 (division by zero). Therefore, x = 0 must be excluded from the domain of f ∘ g.
f(x) Domain: f(x) = 3x is defined for all real numbers.
Checking Options: Let's check each option:
x = -3: g(-3) = -2/3, which is within the domain of f.
x = 0: Excluded due to g(x) being undefined at 0.
x = 1: g(1) = 4, which is within the domain of f.
x = -1: g(-1) = -2, which is within the domain of f.
Conclusion: Only x = -3 and x = -1 satisfy the condition of g(x) being defined and within the domain of f.
Therefore, the values in the domain of f ∘ g are x = -3 and x = -1, making (D) x=-1 the correct answer.