Period of a function

Hey.

I have been given this function:
f(x)=cos(2x+(pi/3))+sin(1.5x-0.25pi)
and asked to find it's period.

I know that it's period is, at a maximum, 4pi by considering periods of the composite functions which make up f(x). However, how can I be sure there is not a smaller value k for which f(x)=f(x+k) for all x? (Short of drawing the graph). For example, g(x)=cosx+sin(x-0.5pi) has a property related to periodicity (it may not be periodic, I do not know) which makes me think I cannot assume it be the lowest common multiple of the two periods pi and 4/3 pi.

Please add anything you can.

When you have f(x)=g(x)+h(x) where both g(x) and h(x) are periodic, the period of f(x) will be the lowest common multiple of the two periods, as this is the smallest number for which both the periods will divide into it a whole number of times, and so the smallest number for which both g(x) and h(x) will return to the same value.

For your function, you need just the lowest common multiple of pi and 4/3 pi.

Does the case g(x)=cosx+sin(x-0.5pi) not contradict this?

I suppose the case where the two functions cancel each other out is an exception to the rule, but this can only occur if they are both perfectly out of phase with each other and both have the same period. Therefore the rule only applies if f(x) itself is actually periodic.

Fair enough. But where does this rule actually come from? Because the previously mentioned case suggested to me I could not just take it for granted that the lowest common multiple of the two functions in question was the desired period.

By the very definition of the lowest common multiple. f(x) can only return to the same part of the cycle (f(x)=f(x+k)) if both g(x) and h(x) return to the same part of the cycle at the same value of k, and this occurs at the first value of k which is a multiple of the periods of both g(x) and h(x). If there was a smaller value at which this occurred, then that would imply that this would be the lowest common multiple instead.

In the case that you mentioned where they both cancel out, then that is a special case as it means that f(x) is not periodic, and thus can't have a period at all, but you could think of it as having a period of 2pi and an amplitude of 0.

"f(x) can only return to the same part of the cycle (f(x)=f(x+k)) if both g(x) and h(x) return to the same part of the cycle at the same value of k"

But what if, if f(x)=g(x)+m(x), g(x)=m(x+k) and m(x)=g(x+k) for some k? I do not know whether this is impossible but I cannot see why it is impossible and so am disposed to consider it. This does not require k to be a multiple of both periods at all (I do not think).

But then wouldn't f(x), g(x) and m(x) all have a period of k, with k being the LCM of k and k?

For example, f(x) = sin(x) + sin(x), Sin(x)=sin(x+2pi), and the period of f(x) is also 2pi, which is the LCM of both periods?

I don't think that's necessary. I do not know the answer to this at all, it just doesn't seem so clear cut.

Have you managed to solve the problem?

Not without just looking at the graph which I cannot do in STEP. But I have solved the rest of the question yes. This is the one bit which I am not so happy with. I see you have an offer too. Nice one!

I'm pretty sure that for any question on STEP that asks about the period of the overall function made from two other periodic functions it will simply be the LCM of the two periods, and it works in this case too. Thanks, what college is your offer from?