Question

$f\left(x\right)=nx+n-[nx+n]+\tan \frac{\pi x}{2},$ where $\left[x\right]$ is the greatest integer $\le x$ and $n\in N$ is

• A. a periodic function of period $1$
• B. a periodic function of period $4$
• C. not periodic
• D. a periodic function of period $2$

Answer 1

Answer: (D)

a periodic function of period $2$

Note that the period of $\tan \left(\frac{\pi x}{2}\right)$ is $\frac{\pi }{\frac{\pi }{2}}=2$. The function $nx+n-[nx+n]=\left\{nx+n\right\}$, where $\left\{x\right\}$ is the fraction part of $x$. which is periodic with period $1$.
Hence, the period of the function $f\left(x\right)$ is the LCM of the two periods which is $2$.