question 7: show that f is bijective

find the inverse function of f.

I need the specific steps for the question 1

 

 

MTH299 - Homework 3 Name: YOUR NAME Homework 3; Due Wednesday, 09/21/2016
Answer-Only Questions. Credit based solely on the answer.
Question 1. Construct truth tables for the following:
(a) not(A and (not B)),
Solution.
A B A and (not B)
T T
T F
F T
F F not(A and (not B)) (b) not(A and B),
(c) (not A) and (not B),
(d) (A and (not B)) or (B and (not A))
Question 2. Negate the following:
(a) A is true and B is false.
(b) A is false or B is true.
(c) A is true and B is true.
(d) A is true or B is true.
Question 3. Construct truth tables for the following:
(a) A or (B and C),
Solution.
A B C
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F MSU B or C A and (B or C) 1 Due: 09/21/2016 MTH299 - Homework 3 Name: YOUR NAME (b) (A or B) and (not C),
Question 4. List all possible different functions f : {a, b, c} → {1, 2}
Solution.
x f1 (x)
a
1
b
1
c
1 Medium-Answer Questions. Provide brief justifications for your responses.
Question 5. Which of the following are statements? Explain.
(a) Pythagorus was friendly.
(b) There exists some number x such that ex = 1 − x2 .
(c) The square root of every positive integer is an irrational number.
(d) Aristotle was Greek.
(e) There are an infinite number of prime numbers.
(f) x2 − x + 1 = 0.
(g) Let n be an integer. Then n is either even or odd.
√
(h) The number 5 is rational.
Question 6. Let F : [2, 1 + e] → [0, 1] be the function defined via the assignment F (x) = ln(x − 1).
(a) Show that F is a bijection.
(b) Compute the function F −1 and state the domain and range of F −1 . MSU 2 Due: 09/21/2016 MTH299 - Homework 3 Name: YOUR NAME Full Proof Questions. Provide complete justifications for your responses.
Question 7. Consider the function f : R2 → R2 defined by f (x, y) = ((x4 + 1)y, x − 1).
1. Show that f is bijective.
2. Find the inverse function of f , and carefully justify that this is the inverse.
Question 8. Consider the function D : P4 → P3 , given by taking the derivative. That is, for a polynomial
df
.
f , it is defined as D(f ) = dx
1. Prove that D is surjective.
2. Prove that D is not injective.
Question 9. Let f : R → R be given by f (x) = ecx for some constant c ∈ R (not zero). Prove, using
induction, that f (n) (x) = cn ecx . (Note that by f (n) , we mean the nth derivative of f .)
You should only use the following facts (it is possible you don’t use all of them) in your argument:
(i) d x
e = ex
dx (ii) the chain rule
(iii) the product rule
(iv) if c is a constant, then d
c = 0.
dx Question 10. Prove that n + 3 < 5n2 for any integer n ≥ 1.
Hint: Use induction.
Question 11. Prove, using induction, that for any r ∈ R with r 6= 1, and any integer n ≥ 0,
n
X
i=0 MSU ri = 1 − rn+1
1−r 3 Due: 09/21/2016

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