Understand first that this is dangerously close to numerology in astronomy. Astronomers wanted for whatever reason to have a very "simple" functional form for the distribution of galaxy luminosities, and so of course if you look close enough (or even not that close as the case may be) real data will never be fit terribly well by this form, especially over a large range of luminosities.
As for the support, it is technically $(0,\infty)$ in luminosity $L$ (OP's variable $x$). Using more common notation I'll write the Schechter function as
$$ \Phi(L) = \frac{n_0}{L^*} \left(\frac{L}{L^*}\right)^\alpha \mathrm{e}^{-L/L^*}, $$
where $\Phi(L)$ measures the number density per unit volume per unit luminosity of galaxies with luminosity $L$ and my $n_0$ is some authors' $\Phi^*$, but this gives $\Phi$ and $\Phi^*$ different units. You quickly see a problem with normalizability depending on $\alpha$. If $\alpha \leq -1$, then
$$ \int_0^{L_\text{max}} \Phi(L)\ \mathrm{d}L $$
diverges for any $L_\text{max}$, indicating infinitely many galaxies (almost all very dim) per unit volume. Fits to real data often do have $\alpha$ hovering near this regime, so the problem is not merely academic.
Of course as pointed out in the original paper by Schechter,
$$ \int_0^\infty L \Phi(L)\ \mathrm{d}L = n_0 L^* \Gamma(\alpha+2) $$
for $\alpha > -2$, so the luminosity density is still finite for reasonable values of $\alpha$.
When fitting data, note that the disagreement is usually bad enough within two or three (astronomical) mags of $L^*$ that these datapoints are often not included in the fit. I suppose a smooth weighting function that de-weights these points according to their proximity to $L^*$ could also be used.
In summary: The normalization is tied to the actual number density of galaxies in the sample, which in some cases can be infinite.
