f(x)=`cosx/([2x/pi]+1/2)` where x is not an integral multiple of `pi` and [ . ] denotes the greatest integer function, is (a)an odd function (b)an even function (c)neither odd nor even (d)none of these

Options

A. an odd function
B. an even function
C. neither odd nor even
D. None of these

Explanation

To determine whether the function \( f(x) = \frac{\cos x}{\left[\frac{2x}{\pi}\right] + \frac{1}{2}} \) is an odd function, even function, or neither, we will follow these steps:

### Step 1: Understand the function
The function is defined as:
\[ f(x) = \frac{\cos x}{\left[\frac{2x}{\pi}\right] + \frac{1}{2}} \]
where \( \left[ . \right] \) denotes the greatest integer function, and \( x \) is not an integral multiple of \( \pi \).
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