Which of the following functions are odd? I. $f(x)=x\left(\frac{e^x-1}{e^x+1}\right)$ II.…

Which of the following functions are odd? I. $f(x)=x\left(\frac{e^x-1}{e^x+1}\right)$ II. $f(x)=k^x+k^{-x}+\cos x$ III. $f(x)=\log \left(x+\sqrt{x^2+1}\right)$
  1. II
  2. I, II
  3. III
  4. I

Solution

III. Since, $f(x)=\log \left(x+\sqrt{x^2+1}\right)$ $\begin{aligned} & \because f(-x)=\log \left(-x+\sqrt{x^2+1}\right) \\ & =-\log \left(\frac{1}{\left(\sqrt{1+x^2}-x\right.} \times \frac{\sqrt{1+x^2}+x}{\left(\sqrt{1+x^2}+x\right)}\right)=-\log \left(x+\sqrt{x^2+1}\right) \\ & \Rightarrow f(-x)=-f(x) \Rightarrow \text { So, } f(x) \text { is odd function. } \end{aligned}$

Asked in: AP EAMCET 2024 (23 May Shift 1)

Practice more Functions questions on Aicharya