Remote work for Tenneesee working from home in Michigan

Hello,

I work from home remote in Michigan, but work for a hospital in Tennessee. Who would I be paying my State withholding tax too? Also, I noticed they are not deducting city tax from my earnings, I dont want to get stuck with a huge city tax bill in the spring... how do I know if I have to pay my resident city income earned tax since I am technically earning my money for work I do in another state?

 

thanks!