periodicity of functions

Question

f(x)={x} 

{x}=x-[x]

Find the periodicity of the functions : 

f(x)={ax} and f(x)=a{x}

Answer 1

Note that if a function f(x) is period p, then

 

af(x) still has period p for a≠0 because you are just rescaling the values of the function, and

 

f(ax) has period p/a because the value of f(x) at x = k is the same as that of f(x) at x = k/a, so the graph is scaled by a factor 1/a along the horizontal direction.

 

Since {x} has period 1, we therefore find that a{x} still has period 1 but {ax} has period 1/a