How to find out the period of fractional part of x

Question

I came across this solved example in a book, it says - Find the period of the function : $f(x)=\sin(4\pi x)+\{3x\}$, where $\{x\}$ denotes the fractional part of $x$.

Now I know that if $f(x)$ is periodic with period $T$, then $f(ax+b)$ is periodic with period $\dfrac{T}{|a|}$.

So, period of $\sin(4\pi x)$ is $\dfrac{2 \pi}{4 \pi} = \dfrac{1}{2}$.

The book says the period of $\{3x\} = 1/3$. Please explain it to me how to find out the period of a fractional part ?

Answer 1

Intuitively, $3x$ increases from $0$ when $x=0$ to $1$ when $x=1/3$. The fractional part then repeats: it is $0$ at $x=1/3$ and increases to $1$ when $x=2/3$

More formally, write $3x=\{3x\}+3\frac n3$ where $n$ is the largest natural such that $\frac n3 \le x$

Answer 2

From the graph of {x}, you can imply that the period is 1. So applying the formula for finding period, we get period = T/|a| =1/(co-efficient of x) = 1/3