HW #9

Reminder: Each HW problem solution should consist of:

• a restatement of the entire problem (you may copy it word for word),
• followed by a complete solution with all intermediate steps shown.

All exercises are from §2.3 (functions) and §2.4 (sequences and summations).

1. (3 points each; total = 9 points)

   p. 146: 12a, b, c

   • You are given some functions and asked to determine which are 1-1.
   • For full credit, you must justify your answers.

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2. (3 points each; total = 9 points)

   pp. 146-147: 14a, c, e

   • You are given some functions and asked to determine which are onto.
   • For full credit, you must justify your answers.

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3. (3 points each; total = 6 points)

   Let R = {x | x is a real number}.
   Let f, g : R → R such that (∀x ∈ R)[f(x) = x³ ∧ g(x) = 2x]

   Compute:

   1. (f o g)(x)
   2. (g o f)(x)

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4. (6 points)

   p. 148: 68.

   • You are asked to prove that a certain kind of function is 1-1 iff it is onto.
   • Hint: Use the definitions of "1-1" and of "onto" together with the fact that:
     if |A| = |B|, then:
     if a function from one of these to the other is not 1-1 or not onto, then |A| ≠ |B|.

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5. (3 points)

   p. 149: 74a

   • You are asked to prove that a partial function (i.e., a function that is not defined on some elements of its domain) can be "extended" to a total function by assigning an arbitrary image to each element for which it is not defined.
   • Hint: All you have to do is show that f * satisfies the definition of a function.

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6. (3 points)

   p. 161: 4a

   • You are asked to compute the first 4 terms of a sequence.

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7. (3 points)

   List the first 10 terms of the sequence {a_n} whose first two terms are a₀ = –3 and a₁ = 2, and which is such that each succeeding term is the sum of the two previous terms.

   • I.e., a_n+2 = a_n + a_n+1

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8. (3 points)

   p. 161: 16a

   • You are asked to compute the value of a summation (a.k.a., a "series"), i.e., to compute the sum of the terms of a sequence.
   • Hint: Compute the terms of the sequence, and then add them up!

Total points = 42.

Tentative grading scheme:

A       41 - 42
A-      38 - 40
B+      36 - 37
B       34 - 35
B-      31 - 33
C+      29 - 30
C       24 - 28
C-      20 - 23
D+      15 - 19
D        8 - 14
F        0 -  7