Math 3210-1
HW 5
Due Monday, June 18, 2007
Functions
1. Find the range of each function f : R → R.
(a) f (x) = x2 + 1.
(b) f (x) = (x + 3)2 − 5.
(c) f (x) = x2 + 4x + 1.
(d) f (x) = 2 cos 3x.
2. Let S be the set of all circles in the plane, and let T be the set of all circles in the plane that are
centered at the origin. Then T ⊂ S.
(a) Define f : S → [0, ∞) by f (C) = the area enclosed by C, for all C ∈ S. Is f injective? Is f
surjective?
(b) Define g : T → [0, ∞) by g(C) = the area enclosed by C, for all C ∈ T . Is g injective? Is g
surjective?
3. Suppose that f : A → B, g : B → C, and h : C → D. Prove that h ◦ (g ◦ f ) = (h ◦ g) ◦ f .
4. In each part, find a function f : N → N that has the desired properties.
(a) surjective, but not injective.
(b) injective, but not surjective.
(c) neither surjective nor injective.
(d) bijective.
5. Find examples to show that equality does not hold in the following theorem: Suppose that f : A → B.
Let C, C1 and C2 be subsets of A and let D, D1 , and D2 be subsets of B. Then the following hold:
(a) C ⊆ f −1 [f (C)].
(b) f [f −1 (D)] ⊆ D.
(c) f (C1 ∩ C2 ) ⊆ f (C1 ) ∩ f (C2 ).
6. Suppose that f : A → B and suppose C ⊆ A and D ⊆ B.
(a) Prove or give a counterexample: f (C) ⊆ D iff C ⊆ f −1 (D).
(b) What condition on f will ensure that f (C) = D iff C = f −1 (D)? You need not prove your answer,
but explain what might go wrong without this condition.
7. Find an example of functions f : A → B and g : B → C such that f and f ◦ g are both injective but g
is not injective.
8. Suppose that g : A → C and h : B → C. Prove that if h is bijective then there exists a function
f : A → B such that g = h ◦ f . Hint: Draw a picture.