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