Switch from the position representation to the momentum representation

Question

If we use Fourier Transform, we can switch from the position representation to the momentum representation, like the following equation

$\phi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int \psi(x)e^{-ipx/\hbar}dx$

here comes the problem, if we use dirac notation we can see it is the inverse Fourier Transform we use to switch into the momentum representation

because of $\langle x|p\rangle=\frac{1}{\sqrt{2\pi \hbar}}e^{\frac{i}{\hbar}px}$,we can get $\begin{split} |p\rangle=&|x\rangle\langle x|p\rangle=\frac{1}{\sqrt{2\pi \hbar}}e^{\frac{i}{\hbar}px}|x\rangle=\mathscr{F} ^{-1}|x\rangle\\ &\mathscr{F} ^{-1}=\mathscr{F} ^{-1}|x\rangle\langle x|=|p\rangle\langle x| \end{split}$

This is used in one published paper. title:" A finite-dimensional quantum model for the stock market" author:Liviu-Adrian Cotfas, page 7, formula 32

also how can I get the following relation satisfied by  the position and momentum operators ? 

$\hat{p}=\mathscr{F} ^{-1} \hat{x} \mathscr{F}$

many thanks!