We can write $g(x)$ as:
$$g(x) = f(x-x_0) = f(x) e^{-i\omega x_0}$$
where $\omega$ is the frequency variable. To see why this is true, we can use the time-shifting property of the Fourier transform:
$$\mathscr{F}\{f(x-x_0)\} = e^{-i\omega x_0} F(\omega)$$
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