First slide

Functions (XII)

Question

Which of the following functions is an odd
function

Moderate

A

$y = x^{4} - 2 x^{2}$
B

$y = x - x^{2}$
C

$y = \cos x$
D

$y = x - \frac{x^{3}}{6} + \frac{x^{5}}{40}$

Solution

If $f (x) = x^{4} - 2 x^{2}$ then $f (- x) = (- x)^{4} - 2 (- x)^{2} =$
$x^{4} - 2 x^{2} = f (x)$ .Hence $f$ is an even function. If $u (x) = x - x^{2}$
then $u (- x) = x - x^{2}$ so $u$ is neither even nor an odd function
$p (x) = cbs x, p (- x) = \cos (- x) = \cos x = p (x)$ so $p$ is even
function. If $s (x) = x - \frac{x^{3}}{6} + \frac{x^{5}}{40}$ then $s (- x) = - x - \frac{x^{3}}{6} - \frac{x^{5}}{40} =$
$- (x - \frac{x^{3}}{6} + \frac{x^{5}}{40}) = - s (x)$ so $s$ is an odd function

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