Question

The function f is periodic. If f(3) = -3, f(5) = 0, f(7) = 3, and the period of the function of f is 6, then how do you find f(135)?

Answer 1

`f(135)=f(3)=-3`

Explanation:

If the period is `6`, it means that the function repeats its values every `6` units.

So, `f(135)=f(135-6)`, because these two values differ for a period. By doing so, you can go back until you find a known value.

So, for example, `120` is `20` periods, and so by cycling `20` times backwards we have that

`f(135)=f(135-120)=f(15)`

Go back a couple of periods again (which means `12` units) to have

`f(15)=f(15-12)=f(3)`, which is the known value `-3`

In fact, going all the way up, you have

`f(3)=-3` as a known value

`f(3)=f(3+6)` because `6` is the period.

Iterating this last point, you have that

`f(3)=f(3+6)=f(3+6+6)=f(3+6+6+6)=...=f(3+132)=f(135)`, since `132=6*22`