Let function f(x) = x^2 + x + x - x + (1 + |x|) be defined over t… - Vidyadip

MCQ

Let function $f(x) = {x^2} + x + \sin x - \cos x + \log (1 + |x|)$ be defined over the interval $[0, 1]$. The odd extensions of $f(x)$ to interval $[-1, 1]$ is

A
${x^2} + x + \sin x + \cos x - \log (1 + |x|)$
✓
$ - {x^2} + x + \sin x + \cos x - \log (1 + |x|)$
C
$ - {x^2} + x + \sin x - \cos x + \log (1 + |x|)$
D
None of these

✓

Answer

Correct option: B.

$ - {x^2} + x + \sin x + \cos x - \log (1 + |x|)$

b
(b) Odd extension from $[0, 1]$ to $[-1, 1]$ means we have to choose the function out of $(a), (b), (c), (d)$ which satisfies the condition $f( - x) = - f(x).$
Now $| - x|\,\, = \,\,|x|$
$f( - x) = {x^2} - x - \sin x - \cos x + \log \,(1 + |x|)$
$ = - \,[{\rm{function given in (b)]}}$
$\therefore \,\,\,({\rm{b}})$ is correct. No other given function satisfies this criteria of $f( - x) = - f(x).$

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