First slide

Introduction to real valued functions

Question

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

Moderate

A

an even function
B

an odd function
C

a periodic function
D

neither odd nor even

Solution

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

$∴ - f (x) = f (- x)$ for all x. Also, none of the pieces of definition are periodic.

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