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!