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In this paper, we first give a characterization of an $f$-biharmonic curve$\gamma:(a,b)\to N^n(C)$ in an $n$-dimensional space form by solving $ODEs$. Wealso give complete classifications of proper $f$-biharmonic curves in a3-dimensional sphere $S^3$ and hyperbolic space $H^3$. We also derive$f$-biharmonic Riemannain submersion equation from 3-manifolds. We then use itto study $f$-biharmonic Riemannain submersions from a 3-space form $M^3(c)$ toa 2-space form $N^2(c)$ with constant sectional curvature $c$, and alsoconstruct many proper $f$-biharmonic Riemannain submersions from$\mathbb{R}^3$, $H^3$, Sol space and a product space$S^2\backslash\{N,S\}\times\mathbb{R}$.

$f$-biharmonic curves in space forms and $f$-biharmonic Riemannain submersions from 3-manifolds