For each of the following, determine whether the function is one-to-one, onto, or both. Prove your assertions. a. f W Z ! Z defined by f .x/ D 2x. b. f W Z ! Z defined by f .x/ D 10 C x. c. f W N ! N defined by f .x/ D 10 C x. d. f W Z ! Z defined by f .x/ D ( x 2 if x is even x1 2 if x is odd: e. f W Q ! Q defined by f .x/ D x 2

Lecture 7: Limits (Section 2.2) Recall the deﬁnition: For a given function f(x), we say that x→c f(x)= L if we can make the values of f(x)aseto L as we want by choosing x suﬃciently close to c on either side but not equal to c. ▯ ex. If f(x)= x if x ▯=1 ,dml f(x). f3i x =1 x→1 ▯ ex. If g(x)= f 3i x ≤ 0 ,ndl g(x). −f1i x> 0 x→0