1. [5 pts] Find fg and gf, where f(x)=x5 and g(x)=x2+3 are functions from R to R. 2. [5 pts] If f and fg are one-to-one, does it follow that g is one-to-one? Prove your answer. 3. [8 pts] Let f be a function from A to B and let S and T be subsets of A (i.e., S,TA ). Prove that (I) f(ST)=f(S)f(T). (II) f(ST)f(S)f(T). 4. [8 pts] Let A,B, and C be three sets such that there exists bijections f:AB and g:AC. Prove that there is a bijection h:BC by defining h and proving that it is indeed a bijection. 5. [7 pts] Let A and B be two finite sets of the same size. Prove that if f:AB is injective, then it is also surjective. 6. [7 pts] Let A and B be two infinite sets such that there exists a bijection between them. Consider an injective function f:AB. Is f necessarily surjective? Prove your