Determine the fourier series of $f(x) = 2x - x^2$ for $0 < x < 3$ and $f(x+3) = f(x)$

Question

The question: Determine the fourier series of f(x)

$f(x) = 2x - x^2$ for $0 < x < 3$

and $f(x+3) = f(x)$

If the questions asks to determine the fourier series of f(x), whats the meaning of including $f(x+3) = f(x)$ ?

What I did was finding the fourier series of $f(x)$ and $f(x+3)$. But, apparently, this is not what the teacher wanted, so I am confused.

Answer 1

The Fourier series is computed over some interval.

The Fourier series, as an approximation to the original function, is only 'valid' in this interval ('valid' refers to convergence).

The Fourier series itself will be periodic with the period given by the length of the interval. Sometimes it is convenient or relevant to define the original function $f$ as periodic over this interval.

If the interval is $[0,3]$ then the periodicity is irrelevant.

If the interval is strictly larger, the the periodicity is part of defining $f$ on the larger interval.

Answer 2

The $f(x+3)=f(x)$ says that the function is periodic with fundamental wavelength $3$. Your first sine wave is then $\sin \frac {2\pi x}3$ and the rest are multiples of that frequency.

Answer 3

The $f(x)=f(x+3)$ means that $f$ is periodic and defined on all real numbers. It was likely mentioned because Fourier series' convergence depends on the continuity of the periodic extensions, although it is not important in the calculation.