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Q.
Which of the following function is an odd function?
Relations and Functions - Part 2
Solution:
(a) $f ( x )=\sqrt{1+ x + x ^{2}}-\sqrt{1- x + x ^{2}}$
$f(-x)=\sqrt{1-x+x^{2}}-\sqrt{1+x+x^{2}}=-f(x)$
$\therefore f ( x )$ is an odd function
(b) $f(x)=x\left(\frac{a^{x}+1}{a^{x}-1}\right)$
$\Rightarrow f(-x)=(-x)\left(\frac{a^{-x}+1}{a^{-x}-1}\right)$
$=(-x)\left(\frac{1+a^{x}}{1-a^{x}}\right)$
$=x\left(\frac{ a ^{x}+1}{ a ^{ x }-1}\right)= f ( x )$
$\therefore $ It is an even function
(c) $f(x)=\log \left(\frac{1-x^{2}}{1+x^{2}}\right) $
$\Rightarrow f(-x)=\log \left(\frac{1-x^{2}}{1+x^{2}}\right)=f(x)$
$\therefore $ It is an even function
(d) $f(x)=k \Rightarrow f(-x)=k=f(x)$
$\therefore $ It is an even function