Proving Periodicity for a Function with Two Elements and a Constant Period

[darkmagic]

Homework Statement

The elements of f(t) are 1+kt and t here are 0<t<1
the other element is 1 and t here are 1<t<2.

f(t+2)=f(t)

prove if k is not equal to zero, the period f is 2.

i know the graph of element 1 but how should I graph the other element with k.
Based from the graph, the period is 2.

How should I prove that using mathematical statement?

Homework Equations

The problem is all about fundamental period

 

[Mark44]

What do you mean "elements of f(t)"?
I don't have any idea of what you saying here:

The elements of f(t) are 1+kt and t here are 0<t<1
the other element is 1 and t here are 1<t<2.

Are you saying that f(t) = 1 + kt, if 0 < t < 1, and f(t) = 1, if 1 < t < 2?
Is f(t + 2) = f(t) given, or is that what you're trying to prove?

 

[darkmagic]

thats what i am saying. f(t+2) =f(t) is given. what I am trying to prove is that if k is not equal to zero, the period f is 2.
how i should i plot the 1+kt?

 

[Mark44]

Well, you don't know k, other than it isn't zero, so you have a line segment with pos. slope if k > 0, and neg. slope if k < 0. It doesn't really matter whether it slopes up or down, just so long as the line isn't horizontal.

 

[darkmagic]

ok so how I prove by mathematical statement that the period f is 2? can this be
or it can only be proved by graph?

 

[Mark44]

A function is periodic with period p iff f(x + p) = f(x) for all x. (See the definition on this Wikipedia page: http://en.wikipedia.org/wiki/Periodic_function.)

So the information that you tell me is given would seem to do the trick except when k = 0. In that case, f(x) = 1 at most points, which would imply that the period was smaller than 2. I say "at most points" because from the information in your post, f(x) is not defined at x = 0, x = 1, and x = 2.

Are you sure you have posted the problem exactly the way it reads? It bothers me that f isn't defined at 0, 1, and 2, and it also bothers me that you are given that f(t + 2) = f(t), since that's essentially what you need to prove.