आख़िरकार  =  how
After all, she is your boss, so invite her. / he is, after all, our president.

आख़िरकार  =  how
So you've come after all! / so you made it after all!

आख़िर  =  how
He should have offered to pay- he has plenty of money, after all. / he should have paid. he suggested it, after all.