First slide

Introduction to real valued functions

Question

$f (x) = \frac{\cos x}{[\frac{2 x}{π}] + \frac{1}{2}},$ where x is not an integral multiple of $π$ and [ ] denotes the greatest integer function is

Moderate

A

an odd function
B

even function
C

neither odd nor even
D

both even and odd

Solution

$f (x) = \frac{\cos (- x)}{[- \frac{2 x}{π}] + \frac{1}{2}} = \frac{\cos x}{- 1 - [\frac{2 x}{π}] + \frac{1}{2}}$
(as x is not an integral multiple of $π$ )
$\Rightarrow f (- x) = - \frac{\cos x}{[\frac{2 x}{π}] + \frac{1}{2}} = - f (x)$
$\Rightarrow f (x)$ is an odd function

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