Can home pregnancy test tell if your pregnant after a week ?

Yes: You want to do the test on your urine first thing in the morning (first morning void). There are some kits that can accurately detect pregnancy the first day of your missed period but almost all of them are good within the first week of pregnancy.

Yes and no: It depends on how sensitive the home pregnancy test is. Otherwise, a blood test for a quantitative beta HCG (test which detects the amount of hormone given by the fertilized egg).