Nope.
Consider this wavefunction from Wikipedia :)
[tex]\psi(x, t) = \begin{cases} \frac{1}{\sqrt{L}} \exp\left[ i(kx - \omega t) \right] & \text{ if } 0 \le x \le L \\ 0 & \text{otherwise} \end{cases}[/tex]
(this is actually the wavefunction of a particle in a 1-D box of size L).
Now you can check that [itex]\psi[/itex] is normalized (by the 1/L^(1/2) in front) and take [itex]f = g = \psi[/itex]
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