Question

How many of the following functions are even [sin x is odd and cosx is even]

(a) f(x) = $x^2|x|$ (b) f(x) = $e^x+e^{-x}$

(c) f(x) = log[$\frac{1-x}{1+x}$] (d) log($\sqrt{x^2+1}$- x)

(e) f(x) = log(x + $\sqrt{x^2+1}$ (f) $a^x-a^{-x}$

(g) f(x) = $\sin x+\cos x$ (h) $\sin x\times (e^x-e^{-x})$

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Answer 1

Here, we will first find f(-x). Then we will check if f(x) = f(-x) for even functions and f(x) = -f(-x) for odd functions.

(1) f(-x) = $(-x{)}^2$|-x|) = $x^2|x|=f\left(x\right)$

$\Rightarrow$ Even function

(2) f(-x) = $e^{-x}+e^{-(-x)}=e^{-x}+e^x=f\left(x\right)$

$\Rightarrow$ Even

(3) f(-x) = log ($\frac{1-(-x)}{1-x}$) = log($\frac{1+x}{1-x}$

= -log($\frac{1-x}{1+x}$ = -f(x)

$\Rightarrow$ Odd function

(4) f(-x) = log($\sqrt{\left({-x}^2\right)+1-(-x)}$)

= log($\sqrt{\left({-x}^2\right)+1}$+ x)

= - log $\frac{1}{(\sqrt{x^2+1}+x)}$= -log($\frac{\sqrt{\left(x^2\right)+1}-x}{\sqrt{\left(x^2\right)+1}+x\sqrt{\left(x^2\right)+1}-x}$)

= -log($\frac{\sqrt{\left(x^2\right)+1}-x}{\sqrt{\left(x^2\right)+1}-x^2}$) = -log$\sqrt{\left(x^2\right)+1}$-x

f(x) = -f(x)

$\Rightarrow$ Odd function

(5) f(-x) = log($\sqrt{\left(x^2\right)+1}$-x)

= log$\frac{\sqrt{\left(x^2\right)+1}-x}{(\sqrt{\left(x^2\right)+1}+x)}$ $\times$ ($\sqrt{\left(x^2\right)+1}$+ x)

= log$\frac{x^2+1-x^2}{\sqrt{\left(x^2\right)+1}+x}$

= log$\frac{1}{(\sqrt{\left(x^2\right)+1}+x)}$

= - log ($\sqrt{\left(x^2\right)+1}$+ x)

(6) f(-x) = $a^{-x}$ - $a^{-(-x)}$

= $a^{-x}$ - $a^x$

= -($a^x$ - $a^{-x}$)

= -f(x)

$\Rightarrow$ Odd function

(7) f(-x) = $\sin$ (-x) + $\cos$ (-x)

= -$\sin$ x + $\cos$ x

=This is not equal to f(x) or f(-x)

$\Rightarrow$ Neither odd nor even

(8) f(-x) = $\sin$ (-x) $\times$ ($e^{-x}$ - $e^{-1(-x)}$)

= -$\sin$ x $\times$ ($e^{-x}$ - $e^x$)

= $\sin$ x ($e^x$ - $e^{-x}$)

= f(x)

$\Rightarrow$ Even function