Sets, Relations & Functions Question - None

Question

Which is an odd function?
(A) $|x-2|+(x+2)\text{sgn}(x+2)$
(B) $\frac{x}{e^x-1}+\frac{1}{2x}$
(C) $\log(\sin x+\sqrt{1+\sin^2 x})$
(D) $e^{-4x}(e^{2x}-1)^4$

Accepted Answer

Key Idea: Use $(\sqrt{1+t^2}+t)(\sqrt{1+t^2}-t)=1$ to show option (C) is odd.

Step 1: Let $L(t)=\log(t+\sqrt{1+t^2})$. Then $L(-t)=-L(t)$ by the identity above.

Step 2: Option (C): $F(x)=L(\sin x)$. Since $\sin(-x)=-\sin x$: $F(-x)=-F(x)$ ✓

Step 3: Option (D) simplifies to $(1-e^{-2x})^4$ -- not odd.

Answer: (C)

Trap: Do not test by complicated expansion. Transform first; the structure becomes obvious.

Key Concept: $\log(t+\sqrt{1+t^2})$ is an odd function

Question:

Step-by-Step Solution

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