Question

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 even and n ≥ 2, F(n) = F(3n−1) if n is odd and n ≥ 3, F(1) = 1.

Answer #1

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 :...

Give a recursive algorithm to solve the following recursive
function.
f(0) = 0;
f(1) = 1;
f(2) = 4;
f(n) = 2 f(n-1) - f(n-2) + 2; n
> 2
Solve f(n) as a function of n using
the methodology used in class for Homogenous Equations. Must solve
for the constants as well as the initial conditions are given.

1. For the following, make sure you explain which basic
functions in F1-F3 you are using, and how exactly you are applying
the operations O1-O3.
(a) Show that every constant function g : N → N is recursive.
You may need to use induction.
(b) Show that the function f : N → N such that f(x) = x 3 is
primitive recursive.

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.

Determine which of the following functions are injective,
surjective, bijective (bijectivejust means both injective and
surjective).
(a)f:Z−→Z, f(n) =n2.
(d)f:R−→R, f(x) = 3x+ 1.
(e)f:Z−→Z, f(x) = 3x+ 1.
(g)f:Z−→Zdefined byf(x) = x^2 if x is even and (x −1)/2 if x is
odd.

QUESTION 1
For the following recursive function, find f(5):
int f(int n)
{
if (n == 0)
return 0;
else
return n * f(n - 1);
}
A.
120
B.
60
C.
1
D.
0
10 points
QUESTION 2
Which of the following statements could describe the general
(recursive) case of a recursive algorithm?
In the following recursive function, which line(s) represent the
general (recursive) case?
void PrintIt(int n ) // line 1
{ // line 2...

***************PLEASE GIVE ANSWERS IN RACKET PROGRAMMING
LANGUAGE ONLY******************
Write a recursive Racket function "update-if" that takes two
functions, f and g, and a list xs as parameters and evaluates to a
list. f will be a function that takes one parameter and evaluates
true or false. g will be a function that takes one parameter and
evaluates to some output. The result of update-if should be a list
of items such that if x is in xs and (f x)...

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.

1. Consider the functions ?(?) = √? + 1 , ?(?) = 2? 4−? , and
?(?) = ? 2 − 5
(a) Find ?(0), ?(0), ?(0)
(b) (??)(?)
(c) (? ∘ ?)(?)
(d) Find the domain of (? ∘ ?)(?)
(e) Find and simplify ?(?+ℎ)−?(?) ℎ .
(f) Determine if ? is an even function, odd function or neither.
Show your work to justify your answer.
2. Sketch the piecewise function. ?(?) = { |? + 2|, ??? ?...

1.Using only the definition of uniform continuity of a function,
show that f(z) = z^2 is uniformly continuous on the disk {z : |z|
< 2}.
2. Describe the image of the circle |z-3| = 1 under the mapping
w = f(z) = 5-2z. Be sure to show that your description is
correct.
Please show full explaination.

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