First slide

Functions (XII)

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

an even function
C

neither odd nor even
D

none of these

Solution

$\begin{array}{l} f (- x) = \frac{\cos (- x)}{[- \frac{2 x}{π}] + \frac{1}{2}} = \frac{\cos x}{- 1 - [\frac{2 x}{π}] + \frac{1}{2}} (As x is not an integral multiple of π) \\ = - \frac{\cos x}{[\frac{2 x}{π}] + \frac{1}{2}} = - f (x) \end{array}$
Therefore, f(x) is an odd function.

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