We study the three-fermion problem within a 1D model of a Feshbach resonance in order to gain insight into how the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like state at small negative scattering lengths evolves into a Bose-Fermi mixture at small positive scattering lengths. The FFLO state possesses an oscillating superfluid correlation function, while in a Bose-Fermi mixture correlations are monotonic. We find that this behavior is already present at the three-body level. We present an exact study of the three-body problem and gain extra insights by considering world lines of a path-integral Monte Carlo calculation. © 2010 The American Physical Society.