The impact of medium-width bands on the selection, and subsequent luminosity function measurements, of high-z galaxies

New, ultra-deep medium-width photometric coverage with JWST's NIRCam instrument provides the potential for much improved photo-z reliability at high redshifts. In this study, we conduct a systematic analysis of the JADES Origins Field, which contains 14 broad- and medium-width near-infrared bands, to assess the benefits of medium band photometry on high-z completeness and contamination rates. Using imaging reaching AB mag $29.8-30.35$ depth, we test how high-z selections differ when images are artificially degraded or bands are removed. In parallel, the same experiments are conducted on simulated catalogues from the JAGUAR semi-analytic model to examine if observations can be replicated. We find sample completeness is high ($80\%+$) and contamination low ($<4\%$) when in the $10\sigma+$ regime, even without the use of any medium-width bands. The addition of medium-width bands leads to increases in completeness ($\sim10\%$) but multiple bands are required to improve contamination rates due to the small redshift ranges over which they probe strong emission lines. Incidents of Balmer-Lyman degeneracy increase in the $5-7\sigma$ regime and this can be replicated in both simulated catalogues and degraded real data. We measure the faint-end of the UV LF at $8.5<z<13.5$, finding high number densities that agree with previous JWST observations. Overall, medium bands are effective at increasing completeness and reducing contamination, but investment in achieving comparable depths in the blue ($<1.5\mu$m) as achieved in the red is also found to be key to fully reducing contamination from high-z samples.