First slide

Functions (XII)

Question

If $f (x) = \sin x - \cos x$ is written as $f_{1} (x) + f_{2} (x)$ where $f_{1} (x)$ is even and $f_{2} (x)$ is odd then

Moderate

A

$f_{1} (x) = \cos x$
B

$f_{1} (x) = - \cos x$
C

$f_{2} (x) = - \sin x + \cos x$
D

$f_{2} (x) = \sin (2 π - x)$

Solution

$f_{1} (x) = \frac{f (x) + f (- x)}{2} = \frac{1}{2} [S i n x - \cos x - \sin x - \cos x] = - \cos x f_{2} (x) = \frac{f (x) - f (- x)}{2} = \frac{1}{2} [\sin x - \cos x + \sin x + \cos x] = \sin x$

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