(1) f(x+1)+f(x+3)=2x, then find the period of the function.

is it 3 by any chance

Suppose the period is T0

Then

f(x+T+1) = f(x+1+T) = f(x+1)

f(x+T+3) = f(x+3+T) = f(x+3)

Hence, f(x+T+1)+f(x+T+3) = f(x+1)+f(x+3) = 2x

However, we also have f(x+T+1)+f(x+T+3) = 2(x+T)

This implies that 2x = 2(x+T) or T = 0

Hence f is non-periodic

f(x+1)+f(x+3)=2x

f(x+1+t)+f(x+3+t)=2x+2t

f(x+1+t)+f(x+3+t)=2t+f(x+1)+f(x+3)

f(x+1+t)+f(x+3+t)=f(x+1)+f(x+3)

so 2t=0.... that means the period doesnt exist since t

i.e.,the function is non-periodic....................