Which of the following functions are odd?I. f | vedclass

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(C) function $f(x)$ is odd if $f(-x) = -f(x)$.$I. f(x) = x \left( \frac{e^x-1}{e^x+1} \right)$$f(-x) = (-x) \left( \frac{e^{-x}-1}{e^{-x}+1} \right) =

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$III$

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(C) function $f(x)$ is odd if $f(-x) = -f(x)$.$I. f(x) = x \left( \frac{e^x-1}{e^x+1} \right)$$f(-x) = (-x) \left( \frac{e^{-x}-1}{e^{-x}+1} \right) = (-x) \left( \frac{\frac{1}{e^x}-1}{\frac{1}{e^x}+1} \right) = (-x) \left( \frac{1-e^x}{1+e^x} \right) = x \left( \frac{e^x-1}{e^x+1} \right) = f(x)$.Since $f(-x) = f(x)$, $I$ is an even function.$II. f(x) = k^x + k^{-x} + \cos x$$f(-x) = k^{-x} + k^x + \cos(-x) = k^{-x} + k^x + \cos x = f(x)$.Since $f(-x) = f(x)$, $II$ is an even function.$III. f(x) = \log \left(x+\sqrt{x^2+1}\right)$$f(-x) = \log \left(-x+\sqrt{(-x)^2+1}\right) = \log \left(\sqrt{x^2+1}-x\right)$.Multiplying and dividing by $\sqrt{x^2+1}+x$:$f(-x) = \log \left( \frac{(\sqrt{x^2+1}-x)(\sqrt{x^2+1}+x)}{\sqrt{x^2+1}+x} \right) = \log \left( \frac{x^2+1-x^2}{\sqrt{x^2+1}+x} \right) = \log \left( \frac{1}{\sqrt{x^2+1}+x} \right) = \log \left( (x+\sqrt{x^2+1})^{-1} \right) = -\log \left( x+\sqrt{x^2+1} \right) = -f(x)$.Since $f(-x) = -f(x)$, $III$ is an odd function.Therefore, only $III$ is an odd function.

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

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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)$