Last Update: 27 March 2009
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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).

- (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.

- (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.

- (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*) =*x*^{3}∧*g*(*x*) = 2*x*]Compute:

- (
*f*o*g*)(*x*) - (
*g*o*f*)(*x*)

- (
- (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|.

- (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.

- (3 points)
p. 161: 4a

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

- (3 points)
List the first 10 terms of the sequence {

*a*} whose first two terms are_{n}*a*_{0}= –3 and*a*_{1}= 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}

- I.e.,
- (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.

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

DUE AT BEGINNING OF LECTURE, FRIDAY, APRIL 3 |

Copyright © 2009 by William J. Rapaport (rapaport@cse.buffalo.edu)

http://www.cse.buffalo.edu/~rapaport/191/S09/hw09.html-20090322