No. Consider polynomials which are also odd functions.
Edit: The above may be useful, but is not correct. If $f_2$ were odd, the answer might be no, but it is unclear that the relations above can ever produce an odd function. End Edit.
Edit2: Thanks to Michael Renardy for producing an odd $f_2$. End Edit2.
Gerhard "Ask Me About System Design" Paseman, 2011.04.12