Questions & Answers

Question

Answers

Answer
Verified

Now, calculating the time period for the function ${\sin ^3}\pi x$ by the concept of trigonometric identity $\sin (2\pi + \theta ) = \sin \theta $ as:

$

{\sin ^3}\pi x = {\sin ^3}\pi (2 + x) \\

= {\sin ^3}(2\pi + \pi x) \\

= {\sin ^3}\pi x \\

$ So, the time period of ${\sin ^3}\pi x$ is 2.

Again, calculating the time period of the function $x - [x]$ by using the concept of an integral part $[x + 1] = [x] + [1] = [x] + 1$ as:

$

x - [x] = x + 1 - [x + 1] \\

= x + 1 - [x] - 1 \\

= x - [x] \\

$ So, the time period of $x - [x]$ is 1.

Now, the overall time period of the given function $f(x) = {2^{{{\sin }^3}\pi x + x - [x]}}$ is the LCM of the time period of the functions ${\sin ^3}\pi x$ and $x - [x]$ which is 2 and 1, respectively.

$

T = LCM(2,1) \\

= 2 \\

$ Hence, the time period of the function $f(x) = {2^{{{\sin }^3}\pi x + x - [x]}}$ is 2.