Answer
See explanation.
Work Step by Step
a. A wave function $\psi$ is normalized when the integral of $|\psi|^2$ over all space is 1. This is because $|\psi|^2dV$ is the probability that the particle is found in the volume dV. The particle must be found somewhere in space, so the integral equals 1.
b. Try to set $P=\int^{\infty}_{-\infty}|\psi|^2 dx=1$.
However, $\int^{\infty}_{-\infty}(e^{ax})^2 dx=\infty$.
This wave function cannot be normalized. It cannot be a valid wave function.
c. We integrate only starting from 0, out to infinity. Start with the normalization condition for this wavefunction.
$P=\int^{\infty}_{0}|\psi|^2 dx=1$
$ A^2\int^{\infty}_{0}e^{-2bx} dx=1$
$ A^2\frac{1}{2b}=1$
$A=\sqrt{2b}$
The units of b are reciprocal meters, so the units of A are $\sqrt{m^{-1}}$.
