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Show that: FOV = f*_{s} / (g G_{f} ) . *
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Assume quadrature sampling of the time domain signal, and adopt the symbol devinitions used in *The Basics of MRI*.
Start with the resonance equation.

n = g B

In the presence of a magnetic field gradient the equation becomes

n = g (B_{o} + x G_{x}).

Relative to the frequency at the isocenter of the magnet, we have

n = n_{o} + g x G_{x}
n - n_{o} = Dn = g x G_{x}

To record a range of frequencies Dn a quadrature sampling rate of f_{s} is required.
In the presence of the magnetic field gradient, this range of frequencies corresponds to the distance across the image in the frequency encoding direction, or the FOV.
Therefore.

f_{s} = g FOV G_{f} ,

or
FOV = f_{s} / (g G_{f}).