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We present a joint analysis of the rest-frame ultraviolet (UV) luminosityfunctions of continuum-selected star-forming galaxies and galaxies dominated byactive galactic nuclei (AGNs) at $z \sim$ 4. These 3,740 $z \sim$ 4 galaxiesare selected from broad-band imaging in nine photometric bands over 18 deg$^2$in the \textit{Spitzer}/HETDEX Exploratory Large Area Survey (SHELA) field. Thelarge area and moderate depth of our survey provide a unique view of theintersection between the bright end of the galaxy UV luminosity function(M$_{AB}<-$22) and the faint end of the AGN UV luminosity function. We do notseparate AGN-dominated galaxies from star-formation-dominated galaxies, butrather fit both luminosity functions simultaneously. These functions are bestfit with a double power-law (DPL) for both the galaxy and AGN components, wherethe galaxy bright-end slope has a power-law index of $-3.80\pm0.10$, and thecorresponding AGN faint-end slope is $\alpha_{AGN} = -1.49^{+0.30}_{-0.21}$. Wecannot rule out a Schechter-like exponential decline for the galaxy UVluminosity function, and in this scenario the AGN luminosity function has asteeper faint-end slope of $-2.08^{+0.18}_{-0.11}$. Comparison of our galaxyluminosity function results with a representative cosmological model of galaxyformation suggests that the molecular gas depletion time must be shorter,implying that star formation is more efficient in bright galaxies at $z=4$ thanat the present day. If the galaxy luminosity function does indeed have apower-law shape at the bright end, the implied ionizing emissivity from AGNs isnot inconsistent with previous observations. However, if the underlying galaxydistribution is Schechter, it implies a significantly higher ionizingemissivity from AGNs at this epoch.

Bridging Star-Forming Galaxy and AGN Ultraviolet Luminosity Functions at $z=4$ with the SHELA Wide-Field Survey