I am travelling to Brazil in a few weeks from the US. I am wondering if I can enter and exit the US on a US passport but use my Swiss passport to enter and exit Brazil. I ask this because Brazil does not require the application for a tourist visa from Swiss citizens, but it does require a visa for US citizens.
Also, will this be a problem upon re-entering the US that I do not have a Brazilian Visa in my US passport. Is the US allowed to know that I am a dual citizen?
Thanks!

PoliticallyCorrect

Of course you can enter Brazil on your Swiss passport.
And US immigration will certainly not look for a Brazilian stamp in your passport.
And since 1986 the US has allowed dual citizenship.

On your exit question; some countries have a separate exit procedure and some do not.
The US does not unless you are crossing a land border.

You should check in for your flight from the US to Brazil using your Swiss passport as this is the one you will travel on. The airline checkin agent will give you a hard time however and will try to force you to show them your US passport as well. They have no legal right to do this as they are airline employees not immigration and you can stand your ground, I often do. I like to wind them up by telling them I am in the country illegally and that they should deport me on the flight I am planning to take as this will save me paying for it. That usually shuts them up!