Math 220
Exam 2 Sample Problems
October 22, 2013
The problems below should give you a sampling of what types of things to expect on the
exam. This is not meant to be an exhaustive list, and there may be types of problems on
the exam that differ from these.
Problems on topics from Chapter 3:
1. Consider the formula f (x) = 2 −
√
x + 4.
(a) What
√ is the largest subset of A ⊆ R so that f : A → R defined by f (x) =
2 − x + 4 is a function?
(b) Compute the image of f : A → R.
(c) Compute f ([5, 12)).
(d) Compute f −1 ([0, 2]).
2. Let f : R → R be defined by f (x) = x4 + x2 .
(a) Compute the image of f . Explain your answer.
(b) Compute f ([−1, 2]). Explain your answer.
3. Determine if the given function is injective, surjective, and/or bijective. Which of
these functions are invertible? No formal proof required.
(a) f : R → [0, 1] defined by f (x) = 1 − sin2 (x).
(b) g : [0, π2 ] → [−1, 1] defined by g(x) = cos(x).
(c) h : Z → 3Z defined by h(n) = 3(n + 1).
(d) j : R × R → [0, ∞) defined by j(x, y) = (x + y + 1)2 .
4. Define f : R − {1} → R − {2} by f (x) =
(a) Show that f is injective.
(b) Show that f is surjective.
(c) Find f −1 (x).
1
2x − 1
.
x−1
5. Let f : A → B and g : B → C be functions. Suppose that f is surjective and that g
is not injective.
(a) Translate “f is surjective” into mathematical symbols.
(b) Translate “g is not injective” into mathematical symbols.
(c) Prove that g ◦ f : A → C is not injective.
6. Let f : A → B be a function. Let W ⊆ B.
(a) Prove that f (f −1 (W )) ⊆ W .
(b) Prove that if f is surjective then f (f −1 (W )) = W .
Problems on topics from Chapter 4: Since the homework due on the day of the exam is
from Chapter 4, studying the assigned problems and others like them from the book would
be good examples.
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