Functions 05

Algebra Level 4

$\large f(x) = \cfrac{\cos x}{\left\lfloor \frac{2x}{\pi} \right\rfloor + \frac{1}{2}}$

What is the value of $f(x)$ above for all $x \ne n \pi$ , where $n \in \mathbb Z$ ?

an odd function none of these neither even nor odd an even function

2 solutions

Raj Rajput
Nov 9, 2015

Tapas Mazumdar
May 22, 2017

$f(x) = \dfrac{\cos x}{\left\lfloor \frac{2x}{\pi} \right\rfloor + \frac 12} \\ \begin{aligned} \implies f(-x) &= \dfrac{\cos (-x)}{\left\lfloor - \frac{2x}{\pi} \right\rfloor + \frac 12} & \small{\color{#3D99F6} \cos(-x) = \cos x} \\ & = \dfrac{\cos x}{-1 - \left\lfloor \frac{2x}{\pi} \right\rfloor + \frac 12} & \small{\color{#3D99F6} \left\lfloor - y \right\rfloor = -1 - \left\lfloor y \right\rfloor \text{ for } y \in \mathbb{R} \setminus \mathbb{Z}} \\ & = \dfrac{\cos x}{- \left\lfloor \frac{2x}{\pi} \right\rfloor - \frac 12} \\ & = - \dfrac{\cos x}{\left\lfloor \frac{2x}{\pi} \right\rfloor + \frac 12} \\ & = - f(x) \end{aligned}$

Thus $f(x)$ is an odd function for any $x$ which is not an integral multiple of $\pi$ .

Hey Tapas, what if it is an odd integer, I mean it is already specified that it can't be an integer but look at this once, $x \ne n \pi$ $\frac{2x}{\pi} \ne 2n$ We had only proved here that it can't be even integer. What if it is odd, then it will be simultaneously odd and even.

Sahil Silare - 4 years ago

Functions 05

2 solutions

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