1. Temperature Readings The equation to convert the temperature from degrees Fahrenheit to degrees Celsius is: c(x) = (x - 32) The equation to convert the temperature from degrees Celsius to degrees Fahrenheit is: f(x) = x + 32 Is there a temperature that has the same reading in both Fahrenheit and Celsius?

2. Function Set of ordered pairs {(x,y)| x  X, y  Y}, where every element of X is associated with a unique element of Y. X is the domain (set of inputs) of the function. Y is the range of the function. The image is the set of outputs.

3. Some Functions to Remember Equal Functions: f(x) = g(x) Identity Function: f(x) = x, id R (7) = 7 Constant Function: f(x) = 3, k(x) = y 0 Absolute Value Function: y = |x|

4. Describing Functions List of ordered pairs Rule Table Graph Function Diagram Verbal Description

5. Examples of Function Rules f(x) = -2x + 1 f(x) = x f(x) = 8 f(x) = x 2 + 7 f(x) = 2 x Associate each integer with a number that is twice the integer.

6. Composite Functions The composition of g with f is the function g o f = g o f(x) = g(f(x)) Notice that g o f is obtained by first doing f and then doing g.

7. Properties of Some Functions One-to-one A function is one-to-one if it never sends two elements of the domain to the same element of the range. Onto A function is onto if no element of the range goes unused.

8. One-to-One or Onto? Temperature functions: c(x) = (x - 32) f(x) = x + 32 f(x) = |x|, Domain = R, Range = R f(x) = x 2, Domain = R, Range = R A function that assigns each word in the English to the first letter in the word. A function that assigns each real number with a point on the number line. y = 2x, Domain = Z, Range = Z

9. Inverses in Mathematics Inverse Property Additive Inverse, Multiplicative Inverse (reciprocal) Inverse Operation Inverse Function If a function has an inverse function, then it is 1-1. If a function is 1-1, then it has an inverse function. g -1 (g(x)) = g (g -1 (x)) = x, or g -1 o g = g o g -1 = id(x)

10. Find the inverse function of each of these functions: y = 2x y = -3x + 5 y = x + 32 y = x 2

11. Solve Using Mental Math Strategies 2  18 11  9 12  13 9  15 90  14 3  36 16  14 8  25 2  15  0 12  1  11

12. Algebra Structures Set Operation(s) with elements in the set Properties that are true but accepted without proof (axioms) Definitions Theorems which can be proved using axioms, definitions and other theorems

13. Field Axioms Associative (+,  ) Identity (+,  ) Inverse (+,  ) Closure (+,  ) Commutative (+,  ) Distributive (  over +)

14. Binary Operation(s) on Set S A binary operation is a function where every combination of two elements of set S results in a unique answer in the set. M: S  S  S For example, addition, subtraction and multiplication with Integers are all binary operations.

15. Sets and Operations Modular Arithmetic: addition, multiplication Set Theory Operations: , , –,  Matrices: addition, multiplication Functions: composition as an operation Symmetries of a Triangle, Rectangle Complex Numbers (a + bi): addition, multiplication

16. The Game of 50 Play with the set of numbers { 1, 2, 3, 4, 5, 6 }. Player 1 chooses a number from the set. Player 2 chooses a number from the set and writes the sum of the two numbers. The players continue choosing numbers and writing sums. The first person to choose a number that results in a sum of 50 wins the game.

17. Properties for mod(n) Activity 4.22 Activity 4.24 Activity 4.25 Activity 4.26 Which of these properties exist for mod(n), using the binary operations + and  ? Commutative, Associative, Identity, (If so, what is the Identity Element?) Inverse

18. Matrices, M 2 (Z) Matrix Addition Matrix Multiplication + =  =

19. Matrix Operations Activity 4.40 - 4.43 (addition) Activity 4.44 - 4.47, 4.48 (multiplication) Which of these properties exist for M 2 (Z), using the binary operations + and  ? Commutative, Associative, Identity, (If so, what is the Identity Element?) Inverse

20. Algebraic Structures Set, Operation(s), Properties Group: A group is a set G together with a binary operation * which satisfy the following: (a) The operation * is associative for all elements of G. (b) G contains a unique identity element, e. If x is any element of g, e * x = x and x * e = x. (c) Each element of G has an inverse in G. If x is any element of g, x -1 is the inverse of x. x * x -1 = e and x -1 * x = e

21. Examples of Groups (Z,+) (Q,+) (R,+) (Q +,  ) (R +,  ) (Z n, + n ) for all n ≥ 1 (M 2 (Z), +)

22. More Algebraic Structures An Abelian Group is a group (G, *) for which the operation is commutative. A Ring is a set R with two operations we will call addition and multiplication, R(+,  ). A ring has the following properties. Associative, Commutative, Identity, Inverse for + (Abelian Group for +) Associative for  Distributive of  over + Examples of Rings: (Z,+,  ), (Q,+,  ), (R,+,  ), (Z n,+,  ), (M 2 (Z),+,  )

23. Fields A field is a set F with two binary operations we will call addition and multiplication, F(+,  ). A Field has the following properties. Associative (+,  ) Commutative (+,  ) Identity (+,  ) Inverse (+, x) (All nonzero elements have an inverse in F.) Distributive (  over +)