ISRO | ISRO CS 2017 | Question 5

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]

Answer: (A)
Explanation: Given, f(x) = log|x| and g(x) = sin(x)

Now,
f(g(x)) = log|g(x)| = log|sin(x)|
So, A = range of log|sin(x)| = (-∞ ,0]

And
g(f(x)) = sin(f(x)) = sin(log|x|)
So, B = range of sin(log|x|) = [-1, 1]

Therefore, A ∩ B = [-1, 0]

Note that -1 ≤ sin(x) ≤ 1 and -∞ ≤ log|x| ≤ ∞

So, option (A) is correct.
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