GE1359
1.
Problem Set 1
Summer 2022
Determine which of the following five sets are equal:
𝐴 = {1, 2, 3}, 𝐵 = {2, 3, 1, 2}, 𝐶 = {3, 1, 3, 2}, 𝐷 = {2, 3, 2, 1}, 𝐸 = {1, 2, 3, 4}
2.
Let 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓}, 𝐵 = {𝑑, 𝑒, 𝑓, 𝑔, ℎ, 𝑖}, 𝐶 = {𝑏, 𝑑, 𝑓, 𝑔}, 𝐷 = {𝑑, 𝑒}, 𝐸 = {𝑒, 𝑓},
𝐹 = {𝑑, 𝑓}, and let 𝑌 be a set which satisfies the following conditions:
𝑌 ⊆ 𝐴, 𝑌 ⊆ 𝐵, 𝑌 ⊈ 𝐶.
Determine which of the sets 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 can equal 𝑌.
3.
4.
Let 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑}, 𝐵 = {𝑏, 𝑑, 𝑓, ℎ} and 𝐶 = {𝑐, 𝑑, 𝑒, 𝑓}. Write down
(a)
𝐴∩𝐵, 𝐴∩𝐶 , 𝐵∩𝐶
(b)
𝐴∪𝐵, 𝐴∪𝐶 , 𝐵∪𝐶
(c)
𝐴\𝐵 , 𝐵\𝐴 , 𝐵\𝐶 , 𝐶\𝐵 , 𝐴\𝐶 , 𝐶\𝐴
Let ℝ be the set of all real numbers, 𝐴 = [1,3] and 𝐵 = (2,4] be two closed intervals.
Write down the following sets:
5.
(a)
𝐴∪𝐵
(b)
𝐴∩𝐵
(c)
(ℝ\𝐴) ∩ 𝐵
(d)
(ℝ\𝐵) ∩ 𝐴
(e)
(ℝ\𝐴) ∩ (ℝ\𝐵)
(f)
(ℝ\𝐴) ∪ (ℝ\𝐵)
(g)
𝐵 ∪ [𝐴 ∩ (ℝ\𝐵)]
(h)
[(ℝ\𝐴) ∩ 𝐵] ∪ [(ℝ\𝐵) ∩ 𝐴]
2
Let 𝑓(𝑥) = 𝑥 3 + 2, 𝑔(𝑥) = 𝑥−1 and ℎ(𝑥) = √𝑥.
Find the formulas for
(a) (𝑓 + 𝑔)(𝑥),
𝑔
(b) (𝑓 ) (𝑥),
(c) (𝑔 ∘ 𝑓)(𝑥),
(d) (𝑓 ∘ 𝑔)(𝑥)
(e) (𝑓 ∘ 𝑔 ∘ ℎ)(𝑥)
and state their largest possible domains.
6.
3
Let 𝐹(𝑥) and 𝐺(𝑥) be two functions defined by 𝐹(𝑥) = 2+𝑥 and 𝐺(𝑥) =
(a)
Find their largest possible domains.
(b)
Find (𝐹 ∘ 𝐺)(𝑥) and state its largest possible domain.
Dr. Emily Chan
1
1+
1
𝑥
.
Page 1
7.
8.
9.
10.
11.
12.
Find the largest possible domain and range for each of the following functions
(a)
𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
(b)
𝑔(𝑥) = 𝑥+2
(c)
ℎ(𝑥) = √𝑥 2 + 4
𝑥−3
Find the largest possible domain for each of the following functions
(a)
𝑓(𝑥) = 5 − 4𝑥 − 𝑥 2
(b)
𝑔(𝑥) = 5−4𝑥−𝑥 2
(c)
ℎ(𝑥) = √5 − 4𝑥 − 𝑥 2
(a)
Show that if 𝑓(𝑥) is an odd function of 𝑥 and 𝑔(𝑥) is an even function of 𝑥, then
(𝑓𝑔)(𝑥) is an odd function of 𝑥.
(b)
Show that if 𝑓(𝑥) and 𝑔(𝑥) are both odd functions of 𝑥, then (𝑓𝑔)(𝑥) is an even
function of 𝑥.
(c)
Show that if 𝑓(𝑥) and 𝑔(𝑥) are both even functions of 𝑥, then (𝑓𝑔)(𝑥) is an even
function of 𝑥.
1
For each of the following functions, determine whether it is an odd function, even function or
neither of them.
sin(𝑥 3 +𝑥)
(a)
𝑓1 (𝑥) =
(b)
𝑓2 (𝑥) = |𝑥 5 + 1|
(c)
𝑓3 (𝑥) = cos3 (2𝑥)
𝑥 4 +3
Let 𝐹(𝑥) be a function defined by 𝐹(𝑥) = 𝑥 2 + 2𝑥 − 3 for 𝑥 ≥ 0.
(a)
Determine whether the function 𝐹(𝑥) is one-to-one and give the range of 𝐹(𝑥).
(b)
Find the inverse function of 𝐹(𝑥) if it exists, then state the domain and range of
𝐹 −1 (𝑥).
For each of the following functions, show that it is one-to-one, then find its inverse function
and state the largest possible domain of the inverse function.
(a)
𝑝(𝑥) = 3𝑥 − 2, 𝑥 ∈ ℝ
(b)
𝑞(𝑥) = 𝑥 2 − 1, 𝑥 ≥ 0
- End -
Dr. Emily Chan
Page 2