Continuity and Differentiability
Properties of Functions
IIT-JEE 2008 Paper 2
Grade 12
Question:
Let $g : (-\infty, \infty) \to (-\pi/2, \pi/2)$ be given by $g(u) = 2 \arctan(e^u) - \pi/2$. Then $g$ is:
(1) even and strictly increasing on $(-\infty, \infty)$
(2) odd and strictly increasing on $(-\infty, \infty)$
(3) even and strictly decreasing
(4) neither even nor odd
Step-by-Step Solution
Key Concept: $g(-u) = 2 \arctan(e^{-u}) - \pi/2$. Now $\arctan(e^u) + \arctan(e^{-u}) = \pi/2$ (complementary identity). So $g(u) + g(-u) = 2[\arctan e^u + \arctan e^{-u}] - \pi = \pi - \pi = 0$. Hence $g$ is odd. $g'(u) = 2e^u/(1 + e^{2u}) > 0$: strictly increasing.
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Correct Answer: (2)