Math 1090
Extra Credit
13 April, 2012
Name:
• This assignment is worth 25 points, added to your test grades. You may
consult your textbook, but you should not consult classmates, friends,
tutors, the internet, or any other resource. This assignment should be
entirely your own work.
• Most of the points will be from parts 4 and 5.
• This assignment is due in class on 23 April.
Part I
2 × 2 Matrices
1. What is the 2 × 2 identity matrix?
I=
2. Compute the following:
2 −3
(a) 1
·I
−1
2
−1 0
(b) I ·
4 8.37
(c)
2π
e2
√
·I
2 log(3)
a b
(d) I ·
c d
3. What do you observe about the identity matrix from these calculations?
4. What does it mean to say that A and B are inverse matrices? (Consult your book if you
aren’t sure.)
5. Are these pairs of matrices inverses? How can you tell? (Show all work! DO NOT
compute the inverse of either matrix)
1 0
1 −2
(a) A =
,B = 4
6 1
−2 5
7
2
1 2
(b) A =
,B =
−3 −1
3 −7
1
π 4π
−4π
π
(c) A =
,B =
0 π1
0
π
1 6 −2
1
6. The two matrices A =
and B = 2
are inverses. Use this information to
−2 1
1 3
solve the equations below:
4
(a) A · X =
−2
π
(b) B · Y =
0
Part II
Functions
7. The identity function is
id(x) = x
(No answer is required to this question.)
8. Let
f (x) = x2 + 6x + 1
g(x) = log(x2 )
h(x) = e4x + x3 − √1x
Compute the following:
(a) (f ◦ id)(x)
(b) (id ◦g)(x)
(c) (h ◦ id)(x)
(d) (id ◦m)(x) where m is any function.
9. What do you observe about the identity function from these calculations?
10. What does it mean to say that f and g are inverse functions? (Consult your book if you
aren’t sure.)
11. Are these pairs of functions inverses? (Show all work! DO NOT compute the inverse of
either function.)
(a) f (x) = 2x + 3, g(x) = 12 x − 3
(b) f (x) = x3 − 3, g(x) =
(c) f (x) = ln
x
3
+ 7, g(x) = 3ex−7
12. The functions f (x) =
the equations below:
(a)
2a+3
5
=5
(b)
5b−3
2
= 16
√
3
x+3
2x+3
5
and g(x) =
5x−3
2
are inverses. Use this information to solve
Part III
Multiplication
13. We call 1 the multiplicative identity.
(No answer is required to this question).
14. Compute the following:
(a) 1 · 6
(b)
√
2·1
(c) 1 · π
(d) x · 1 where x is any real number.
15. What do you observe about the multiplicative identity from these calculations?
16. Two numbers a and b are multiplicative inverses if a · b = 1
(No answer is required to this question)
17. Are these pairs of numbers multiplicative inverses?
(a) a = 3, b = − 31
(b) a = 12 , b = 2
(c) a = −1, b = −1
(d) a = 2, b = −2
18. Use multiplicative inverses to solve the following equations:
(a) 4x = 12
(b) − 31 x = 4
Part IV
Generalisations
19. Thinking about the previous examples, in your own words, what is an identity?
20. In your own words, what is an inverse?
21. In your own words, how are identities and inverses used to solve equations?
Part V
Addition
22. What do you think is meant by additive identity? Explain your answer.
23. When are two numbers additive inverses? Explain your answer.
24. Give three pairs of numbers that are additive inverses.
25. Write two equations that can be solved with additive inverses, and show how to solve
them. (These equations will probably be very simple.)