The function f(x) = x – [x] has period of
Answers
Answer:
F(x) = F(x + P) for all x where P is a positive constant. We can easily check whether the given function
F(x)= x-[x] satisfies the criteria for some positive value of P.
Thus for periodic function
F(x) = x-[x] = x-n where n is an integer ≤ x (1)
F(x + P) = x+ P-[x + P]
Now [x + P] is either ‘n’ or ‘(n+1)’ depending on value of P
If [x + P] is ‘n’ then
F(x + P) = x + P-n (2)
If [x + P]= (n+1) then
F(x + P)= x+P-n-1 (3)
From (1) and (2) for periodicity we must have
X - n= x + P -n i.e. P=0 which is clearly not a solution
From (1) and (3)
X - n = x+P-n-1 → P=1
Thus if we set P=1 it is periodic function with period equal to 1.
Before proceeding analytically; if you try to plot few points on graph manually or by use of excel. we can draw a table and chart for some domain of x and then interpret it generally (x on X-axis and f(x) on y axis.
F(x) = x-[x]
x F(x)
0 0
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5
0.8 0.8
0.9 0.9
1 0
1.1 0.1
1.2 0.2
1.9 0.9
2 0
2.1 0.1
2.9 0.9
3 0