Question

Let $f:Z\to Z$ be defined as $f\left(x\right)=2\sin \left(2\pi x\right)-10\tan \left(5\pi x\right)+7\cos \left(4\pi x\right)+3.$ Then which of the following statements is/are TRUE?

• A. $f\left(x\right)$ is a periodic function
• B. $f\left(x\right)$ is an even function
• C. $f\left(x\right)$ is an odd function and its inverse exists
• D. $f\left(f\right(f\left(x\right)\left)\right)=f\left(f\right(x\left)\right)$ for all $x\in Z$

Answer 1

Answer: (A), (B), (D)

$f\left(f\right(f\left(x\right)\left)\right)=f\left(f\right(x\left)\right)$ for all $x\in Z$

$f:Z\to Z$
$f\left(x\right)=2\sin \left(2\pi x\right)-10\tan \left(5\pi x\right)+7\cos \left(4\pi x\right)+3$
Clearly, $f\left(x\right)=0+0+7+3=10,\forall x\in Z$
Graph of $f\left(x\right)$ is as below :


$f\left(x\right)$ is periodic.
$f\left(x\right)$ is an even function.
$f\left(f\right(f\left(x\right)\left)\right)=10$
Also, $f\left(f\right(x\left)\right)=10$