Take a multimeter or test light and check for power going to each back-up bulb. For a manual transmission, check the back-up light switch on the transmission housing. On an automatic transmission, check the A/T gear position switch (back-up light switch). Back-up circuit flow is from battery to the under-hood fuse/relay box to the ignition switch to the under-dash fuse/relay box to the back-up light switch to the back-up lights. Pretty straight-forward stuff and shouldn't be too hard to diagnose. Focus on the back-up light switch.
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