Typically during a home inspection the water is only tested to see if it is safe to drink. You could end up with a skunk of a well, high in sulfates and iron making it virtually non potable or unable to wash your white clothes without rust staining them but still "safe" to consume.
I'm not sure if the home inspector can get further tests done, maybe the local ag department can help. What ever you do don't depend on one of the local water filter stores like Culigan because they don't have a clue beyond trying to sell you chemicals or a filter to fix a problem that may or may not exist. .
I ended up with a skunk well at a previous house and ended up installing a 2k air charge water system so we could at least shower without smelling rotten egg sulfur, but still couldn't drink the crap.
The well is in a really shallow pressurized aquifer, depending on rainfall, how much water we use, orbit of the moon, my horoscope and the stock market it will smell one day and then nothing the next day..