So I was teaching Math 309 last quarter. The second part of the course is introduction to PDEs. Students are normally given the following problem:
1.One differential equation (say, for example, the standard heat equation) on a rod placed between 0<x<L, t>0
2.Two boundary conditions
3.One initial condition
The last step is to treat the problem as an initial value problem, and find the arbitrary coefficients with the idea of Fourier series.
The Fourier series, by the definition in the textbook we use, should be defined on an interval symmetric with respect to the y-axis. This means, the initial condition needs to get extended, since it was originally defined on only half of the required interval (0<x<L).
A choice is obvious, since the final superposition would be in either sine or cosine series, students should choose to extend the function to be either even or odd to the left of y-axis.
However, when introducing the Fourier series for an arbitrary function, it might be useful to point out that any function (given the appropriate domain of course) can be written as the sum of an even and an odd function. The Fourier series then would naturally be separated into two pieces: even series and odd series. The calculation showing that for odd function, Fourier series only has sine terms is useful.
As far as I have observed, many students do not see the point of extension. Sometimes I would give them a function defined on the right interval (-L<x<L), yet they would do extension anyways.