Let f(x) = log|x| and g(x) = sin x . If A is the range of f(g(x)) and B is the…
2017
Let f(x) = log|x| and g(x) = sin x . If A is the range of f(g(x)) and B is the range of g(f(x)) then A ∩ B is
- A.
[-1, 0]
- B.
[-1, 0)
- C.
[-∞, 0]
- D.
[-∞,1]
Attempted by 16 students.
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Correct answer: A
First, analyze the range of A = f(g(x)). Since g(x) = sin x, |sin x| ∈ (0, 1] where defined. Thus f(g(x)) = log|sin x| ∈ (-∞, 0]. So A = (-∞, 0].
Next, analyze the range of B = g(f(x)). Since f(x) = log|x| spans all real numbers, sin(log|x|) covers [-1, 1]. So B = [-1, 1].
Finally, compute the intersection A ∩ B. The overlap between (-∞, 0] and [-1, 1] is [-1, 0].