Final answer:
Sketch the even and odd extensions of a function over three periods for an even extension g(x) and an odd extension h(x), followed by finding the Fourier cosine and sine series respective to each extension's symmetry.
Step-by-step explanation:
To sketch the graphs of the even and odd extensions of a given function f(x) with period 4, we span the function over three periods while ensuring symmetry about the y-axis for the even extension g(x), and symmetry about the origin for the odd extension h(x).
The Fourier cosine series is determined by projecting f(x) onto the cosines that serve as the basis for even extensions, while the Fourier sine series is determined by projecting f(x) onto the sines that serve as the basis for odd extensions.
For the even extension g(x), the function is mirrored around the y-axis. An even function typically has a cosine Fourier series due to its symmetry about the y-axis. On the other hand, the odd extension h(x) repeats the function across the origin with a change in sign for the negative x-domain. An odd function's Fourier series is typically composed solely of sine terms due to its point symmetry about the origin.