First slide

Functions (XII)

Question

$If f (x) = \{\begin{array}{cl} x^{2} \sin \frac{π x}{2}, & | x | < 1 \\ x | x |, & | x | \geq 1 \end{array}, then f (x) is$

Moderate

A

an even function
B

an odd function
C

a periodic function
D

none of these

Solution

$\begin{array}{l} f (- x) = \{\begin{cases} (- x)^{2} \sin \frac{π (- x)}{2}, & | - x | < 1 \\ (- x) | - x |, & | - x | \geq 1 \end{cases} \\ = \{\begin{cases} - x^{2} \sin \frac{π x}{2}, & | x | < 1 \\ - x | x |, & | x | \geq 1 \end{cases} \\ = - f (x) \end{array}$

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