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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Functions

NDA 2022 · Mathematics · Functions

If f(x)= (x+ 1+x^2 ), then which one of the following is correct?

If f(x)=ln⁡\lleft(x+1+x2),thenwhichoneofthefollowingiscorrect?f(x)=\ln \lleft(x+\sqrt{1+x^2}), then which one of the following is correct?

Options

  1. Af(x)+f(−x)=0f(x)+f(-x)=0
  2. Bf(x)−f(−x)=0f(x)-f(-x)=0
  3. C2f(x)=2f(−x)2 f(x)=2 f(-x)
  4. Df(x)=2f(−x)f(x)=2 f(-x)

Correct answer

A. f(x)+f(−x)=0f(x)+f(-x)=0

Step-by-step solution

∵f(x)=ln⁡(x+1+x2)...(\because \quad f(x)=\ln \left(x+\sqrt{1+x^2}\right) ...( i) f(−x)=ln⁡\lleft(−x+1+x2)...(f(-x)=\ln \lleft(-x+\sqrt{1+x^2}) ...( ii) Adding equations (i) and (ii) f(x)+f(−x)=ln⁡(1+x2+x)+ln⁡(1+x2−x)=ln⁡1=0\begin{aligned} & f(x)+f(-x)=\ln \left(\sqrt{1+x^2}+x\right)+\ln \left(\sqrt{1+x^2}-x\right) \\ & =\ln 1=0\end{aligned}
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