First slide

Functions (XII)

Question

Let the function f (x) = 3 sin x – 4 cos x + log (|x| + $\sqrt{1 + x^{2}})$ be defined on the interval [0, 1]. The odd extension of f (x) to the interval [– 1, 1] is

Moderate

A

$3 \sin x - 4 \cos x + \log (| x | + \sqrt{1 + x^{2}})$
B

$3 \sin x + 4 \cos x - \log (| x | + \sqrt{1 + x^{2}})$
C

$3 \sin x + 4 \cos x + \log (| x | + \sqrt{1 + x^{2}})$
D

None of these

Solution

To make f (x) an odd function in the interval [–1, 1], we re-define f (x) as follows :
$f (x) = \{\begin{array}{lr} f (x), 0 \leq x \leq 1 \\ - f (- x), - 1 \leq x < 0 \end{array}\}$
$= \{\begin{array}{lr} 3 \sin x - 4 \cos x + \log (| x | + \sqrt{1 + x^{2}}), 0 \leq x \leq 1 \\ - [- 3 \sin x - 4 \cos x + \log (| x | + \sqrt{1 + x^{2}})], - 1 \leq x < 0 \end{array}\}$
$= \{\begin{array}{cc} 3 \sin x - 4 \cos x + \log (| x | + \sqrt{1 + x^{2}}), & 0 \leq x \leq 1 \\ 3 \sin x + 4 \cos x - \log (| x | + \sqrt{1 + x^{2}}), & - 1 \leq x < 0 \end{array}\}$
Thus, the odd extension of f (x) to the interval [–1, 1] is
$3 \sin x + 4 \cos x - \log (| x | + \sqrt{1 + x^{2}})$

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