Periodic Functions

State whether the following statements are true/false:

State whether the following statement is true/false:

State whether the following statement is true/false:

State whether the following statement is true/false:

The fundamental period of $f(x)=\cos x$ is $2 \pi$. So, the fundamental period of $g(x)=\cos \frac{7 x}{3}$ is $\frac{6 \pi}{7}$.

State whether the following statements are true/false:

[ix] The fundamental period of $f(x)=\cos \frac{x}{3}$ is $6 \pi$.

State whether the following statements are true/false:

[viii] A constant function is periodic, but it has no fundamental period.

State whether the following statements are true/false:

[vii] If $p$ is a period of a function $f$ with domain D, then $2 p, 3 p, \ldots$ are all periods of $f$.

State whether the following statements are true/false:

[vi] A function $f$ is called periodic if there exists a number $p$ such that $(x+p)$ is in the domain of $f$ whenever $x$ is in the domain, an $f(x+p)=f(x)$ for all $x$ in the domain of $f$.

Find the fundamental period of the function $f(x)=3 \sin \frac{\pi x}{3}+4 \cos \frac{\pi x}{4}$
 

Find the fundamental period of the function $f(x)=4 \cos (2 x+3)$

The period of $f(x)=\sin 2 \sqrt{x+1}$ is:-

The fundamental period of $f\left(x\right)=\sin x$ is $2\mathrm{\pi }$. So the fundamental period of $g\left(x\right)=\sin \frac{x}{6}$ is:-

An odd function is symmetric about the vertical line $x=a \left(a>0\right)$ and if $\sum _{r=0}^{\infty }{\left[f\right(1+4ar\left)\right]}^r=8$, then find the numerical value of $8f\left(1\right)$ $($where $0<f\left(1\right)<1)$.

The period of the function $f\left(x\right)$, which satisfies the relation $f\left(x\right)+f\left(x+4\right)=f\left(x+2\right)+f\left(x+6\right)$ is

Fundamental period of the function $f\left(x\right)={\sin }^{2}4x+\left|\cos 16x\right|$ is

If range of the function $f\left(x\right)={\cos }^2\left(\cos x\right)+{\sin }^2\left(\sin x\right),$ is $[a, b]$ then.

Which of the following functions is/are periodic ?

If the periods of the periodic functions $\mathrm{s}\mathrm{i}\mathrm{n}ax+\mathrm{c}\mathrm{o}\mathrm{s}ax$ and $\left|\mathrm{s}\mathrm{i}\mathrm{n}x\right|+\left|\mathrm{c}\mathrm{o}\mathrm{s}x\right|$ are equal, then $a$ is equal to

The period of the function $f\left(x\right)=\frac{\sin 8x\cos x-\sin 6x\cos 3x}{\cos 2x\cos x-\sin 3x\sin 4x}$ is