Fatty acid breakdown happens in the cytosol of prokaryotes in plants in peroxisomes and in the mitochondrial matrix of all other eukaryotes.  By entering the mitochondrial matrix, the fatty acid is activated by creating a thioester link with CoA. This reaction is catalyzed by acyl CoA synthase (also known as fatty acid thiokinase) which is there on the outer mitochondrial membrane, and uses a molecule of ATP. The entire reaction is irreversible due to the subsequent hydrolysis of PPi to two molecules of Pi.