Question

The function $f\left(x\right)=x-\left[x\right]+\cos x,$ where $\left[x\right]=$the greatest integer less than or equal to $x$, is a

• A. periodic function of in determined period
• B. periodic function of period
• C. non periodic function
• D. periodic function of period 1

Answer 1

The correct option is C non periodic function

note that,$x-\left[x\right]=\left\{x\right\}$ is a periodic function with period 1. Similarly the period of $cos\left(x\right)$ is $2\pi$.
Hence the period of the function $\left(x\right)=x-\left[x\right]+\cos \left(x\right)$ is least common integral multiple of the two period. But since $\pi$ is transcendental number, no such common multiple exist. Hence the function is not periodic.