I think that in order to have a good discussion it would help if you could explain why public transport is slower in your case.

Is the route longer because there isn't a line that goes straight from your home to your workplace?

Is it because there aren't a lot of trains, so they take too long to arrive?

Otherwise it's hard to tell what the problem with public transport in your city.

I live in London for instance, where cars are _almost always_ the slowest option for trips around zones 1-2 or even 3, because the center is always very busy. So when commuting here the fastest option is usually cycling, then 5-10 minutes slower is public transport and driving, which depending on the time of day will also be slower...