Question

If $f\left(x\right)=2^{{\sin }^3\pi x+x-\left[x\right]}$, where $\left[x\right]$ denotes the integral part of $x$ is a periodic function with period

Answer 1

Given $f\left(x\right)=2^{{\sin }^3\pi x+x-\left[x\right]}$.

Now we have $x-\left[x\right]$ is periodic function of period $1$ and ${\sin }^3\pi x$ is periodic function of period $2$ as $x+1-[x+1]=x+1-\left[x\right]-1=x-\left[x\right]$ and ${\sin }^3\pi (2+x)={\sin }^3(2\pi +\pi x)={\sin }^3\pi x$.

Then ${\sin }^3\pi x+x-\left[x\right]$ will be periodic and of period LCM of $1$ and $2$ .i.e. $2$

Since ${\sin }^3\pi x+x-\left[x\right]$ is periodic function of period $2$ then $f\left(x\right)=2^{{\sin }^3\pi x+x-\left[x\right]}$ is periodic of period $2$.

As $f(x+2)=2^{{\sin }^3\pi (x+2)+(x+2)-[x+2]}$$=2^{{\sin }^3\pi x+x-\left[x\right]}=f\left(x\right)$.