Should lemongrass be edible or just a flavoring?

In my limited experience with lemongrass (which I buy fresh-ish in plastic bags from the fridge of my local asian supermarket - not frozen or dried or anything), it's hard and impossible to chew, no matter how long you cook it for. I usually smash it, split it down the middle and add it to soups or other dishes with a lot of liquid, then take it out at the end like a bay-leaf.

However, I've recently seen recipes in which finely chopped lemongrass is put straight in a stir-fry or a sauce, which just seems like a good way to end up picking bits out of your mouth to me. Is there a problem with the lemongrass I'm buying? Is it that only the green is hard and inedible? Or am I cooking it wrong?