Reading Quiz Section 4.3
1. The range of a function f : A → B is (select all that apply):
(a) A subset of the domain. (b) A subset of the codomain.
(c) Always equal to the codomain. (d) Also called the image of the function.
(e) Equal to f (A).
2. Suppose f : A → B and g : B → C are functions. If g ◦ f is bijective, which of the following
must be true?
(a) f is injective. (b) g is injective.
(c) f is surjective. (d) g is surjective.
3. True or False: We can always make a function surjective by making its domain smaller.
4. True or False: If A ⊆ B, there is an injective function f : A → B.
Practice Problems Section 4.3
1. (a) Explain why the ‘rule’ g : {all lines in the plane} → R which sends a line ℓ to the slope of
ℓ does not define a function.
Video Solution
(b) Let L be the set of all non-vertical lines in the plane. The rule f : L → R sending ℓ to its
slope is a well-defined function.
i. Find f (Z) where Z is the set of lines intersecting y = 2x + 5 at exactly one point.
Video Solution
ii. Let U = {−2}. Describe the inverse image f
−1
(U).
Video Solution
iii. Explain why f is not bijective. Find a subset B ⊆ L so that f : B → R is a bijection.
Video Solution 1 Video Solution 2
2. Suppose f : A → B and g : B → C are functions. For each of the following, either find an
example or explain why no such example exists.
(a) f surjective, g not surjective and g ◦ f surjective.
(b) f not surjective, g surjective and g ◦ f surjective.
(c) f surjective, g surjective and g ◦ f not surjective.
(d) f injective, g not injective and g ◦ f injective.
(e) f not injective, g injective and g ◦ f injective.
(f) f injective, g injective and g ◦ f not injective.
Video Solution (parts (a)-(c))
3. Suppose f : A → B is a function. Prove or disprove the following statements:
(a) Let X and Y be subsets of A. If X ∩ Y = ∅ then f (X) ∩ f (Y) = ∅.
(b) Let W and Z be subsets of B. If W ∩ Z = ∅ then f
−1
(W) ∩ f
−1
(Z) = ∅.
Video Solution
