Yup, this is homework. So, here are my answers and thoughts. Am I correct, incorrect or in the wrong ballpark ;)
Determine if the following function is periodic and if so, find the fundamental frequency. (The use of "j" is for imaginary numbers)
1) x(t) = 3cos(4t + pi/3)
This function is periodic (the amplitude doesn't matter..?..).
The fundamental frequency is T = pi/2
2) x(t) = exp( j (pi * t - 1))
From euler form ( exp[x] = cos x - jsin x )
x(t) = cos(pi t - 1) + j sin(pi t - 1)
I do not think this is periodic due to the complex portion.
3) x(t) = [ cos(2t - pi/3) ] ^ 2
Using a Half-Angle identity:
x(t) = .5 [ 1 + cos( 2 ( 2t - pi/3 ) ) ]
= .5 [ 1 + cos(4t - 2pi/3) ]
Period with fundamental period T = pi/2
4) x(t) = Even { cos(4pt*t)) } u(t)
This is not periodic due to the unit step function (as the amplitude changes at t=0)
2) with the complex portion, is x(t) = x(t+T) for some T!=0?
3) correct
4) I'm not sure how you define Even{} or u(t), but cos(4pt*t) is