Question

1. A function f : Z → Z is defined by f(n) = 3n − 9.

(a) Determine f(C), where C is the set of odd integers.

(b) Determine f^−1 (D), where D = {6k : k ∈ Z}.

2. Two functions f : Z → Z and g : Z → Z are defined by f(n) = 2n^ 2+1 and g(n) = 1 − 2n. Find a formula for the function f ◦ g.

3. A function f : R − { 1/2 } → R − {0} is defined by

f(x) = 3/(2x − 1) .

Prove that f is bijective.

Answer #1

3. For each of the piecewise-defined functions f, (i) determine
whether f is 1-1; (ii) determine whether f is onto. Prove your
answers.
(a) f : R → R by f(x) = x^2 if x ≥ 0, 2x if x < 0.
(b) f : Z → Z by f(n) = n + 1 if n is even, 2n if n is odd.

3. Let N denote the nonnegative integers, and Z denote the
integers. Define the function g : N→Z defined by g(k) = k/2 for
even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a
bijection.
(a) Prove that g is a function.
(b) Prove that g is an injection
. (c) Prove that g is a surjection.

Please note n's are superscripted.
(a) Use mathematical induction to prove that 2n+1 +
3n+1 ≤ 2 · 4n for all integers n ≥ 3.
(b) Let f(n) = 2n+1 + 3n+1 and g(n) =
4n. Using the inequality from part (a) prove that f(n) =
O(g(n)). You need to give a rigorous proof derived directly from
the definition of O-notation, without using any theorems from
class. (First, give a complete statement of the definition. Next,
show how f(n) =...

Consider function f (n) = 3n^2 + 9n + 554.
Prove f(n) = O(n^2)
Prove that f(n) = O(n^3)

Let f: Z→Z be the functon defined by f(x)=x+1. Prove that f is a
permutation of the set of integers. Let g be the permutation (1 2 4
8 16 32). Compute fgf−1.

Let f, g : Z → Z be defined as follows:
? f(x) = {x + 1 if x is odd; x - 1 if x is even}, g(x) = {x - 1
if x is odd; x + 1 if x is even}.
Describe the functions fg and gf. Then compute the orders of f,
g, fg, and gf.

Which of the following proposed recursive deﬁnitions of a
function on Z+ produce well-deﬁned functions? If it is well-deﬁned,
provide a direct formula for F(n). Be sure to explain your answer,
either why F is not well-deﬁned, or why your direct formula is
correct. (a) F(n) = 1 + F(bn/2c), for n ≥ 1, F(1) = 1 (b) F(n) =
2F(n−2) if n ≥ 3, F(2) = 1, F(1) = 0 (c) F(n) = 1 + F(n/2) if n is...

Let A =
3
1
0
2
Prove An =
3n
3n-2n
0
2n
for all n ∈ N

Is the function f : R → R defined by f(x) = x 3 − x injective,
surjective, bijective or none of these?
Thank you!

Part II
True or false:
a. A surjective function defined in a finite set X
over the same set X is also BIJECTIVE.
b. All surjective functions are also
injective functions
c. The relation R = {(a, a), (e, e), (i, i), (o, o), (u, u)} is
a function of V in V if
V = {a, e, i, o, u}.
d. The relation in which each student is assigned their age is a function.
e. A bijective function defined...

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