MATH1061/7861, Thu 10 Sep 2020
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Question 2. Let f :Z→Zbe the function f (x) = (x + 1)2.
◦Whatistheimageof1?
◦ What is the image of {1}?
◦ What is the image of {1,2}?
◦ What is the domain? II
◦ What is the codomain?
◦ What is the range?
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◦ What is the preimage of 1?
◦ What is the preimage of 2?
◦ What is the preimage of {0,1,2}?
Question 3. Which of the following are functions?
◦ f :R→Rwhere f (x) = x2
◦ f :N→Nwhere f (n) = n!
◦ f :N→Nwhere f (n) = (n + 1)(n + 3)/3
◦ f :N→Nwhere f (n) = (n + 1)(n + 2)(n + 3)/6
◦ f :R→Rwhere f (x) = 1
◦ f :Q→Zwhere for all m,n ∈Zwith n ≠ 0, f (m/n) = m
◦ f :R×R→Rwhere f ((x,y)) = |x − y| ◦
◦ f : R → R where f = {(x,y) : 3y + 7 = x} ◦
◦ f : R → R where f = {(x,y) : y2 = x} ◦
◦ f :R→Rwhere f = {(x,y) : |x| = |y|}
Challenge. Find all functions f :R→Rwith the property that f (x) + x f (1 − x) = 1 + x2 for all x ∈R.
Question 4. Let f : {1,2,3,4} → {a,b,c} be the function f = {(1,a), (2,a), (3,b), (4,b)}. ◦ Does the inverse function f −1 exist?
◦ What is the preimage of the set {a}?
◦ Is f −1(a) defined? If so, what is it?
◦ Is f −1({c}) defined? If so, what is it?
Question 5. For each statement, either prove it or find a counterexample.
◦ For all functions f : X → Y and g : Y → Z, if g∘f is onto, then f is onto.
◦ For all functions f : X → Y and g : Y → Z, if g∘f is onto, then g is onto.
◦ For all functions f : X → Y and g : Y → Z, if g∘f is one-to-one, then f is one-to-one. ◦ For all functions f : X → Y and g : Y → Z, if g∘f is one-to-one, then g is one-to-one.