Can I give my friend the sour dough "throw away" as a starter to their sourdough starter?
Sure, you can begin a new sour dough starter with the discard from a feeding. However, the reason for discarding isn't simply to reduce the amount.  As your starter matures it also becomes much more acidic.  Acidity is problematic for yeast and bacterial activity and, ultimately, the rise and flavor of your final product.  So, you discard during feeding time to keep the acidity in check, and ensure that your starter is as hospitable to yeast and bacteria as possible.  Your friend can get his or her starter going with your discard, but will then want to get into the habit of discarding when they feed.