Fourier transform output in Mathematica not matching expected result

[Selectron09]

TL;DR

I am attempting to do a problem involving wave functions and imaginary numbers

I am attempting to be able to do this problem with the help of Mathematica and Fourier transform. My professor gave us instructions for Fourier Transformation and Inverse Fourier, but I don't believe that my output in Mathematica is correct.

 

[vela]

You're supposed to define a function of $k$, not $x$, and find the inverse Fourier transform to get $f(x)$. Also, note that $b$ is multiplied by $k$, not $\lvert k \rvert$.

 

[Selectron09]

You're supposed to define a function of $k$, not $x$, and find the inverse Fourier transform to get $f(x)$. Also, note that $b$ is multiplied by $k$, not $\lvert k \rvert$.

You're supposed to define a function of $k$, not $x$, and find the inverse Fourier transform to get $f(x)$. Also, note that $b$ is multiplied by $k$, not $\lvert k \rvert$.

thanks so much!