1. SECTION 1.1 FUNCTIONS FUNCTIONS

2. DEFINITION OF A RELATION A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, we say that x corresponds to y or that y depends upon x. The correspondence can be written as an ordered pair (x,y).

3. DEFINITION OF A RELATION Thus, a relation is simply a set of ordered pairs or a table which relates x and y values.

4. DEFINITION OF FUNCTION Let X and Y be two nonempty sets of real numbers. A function from X into Y is a rule or a correspondence that associates with each element of X a unique element of Y. This is a special type of relation. For every x, there is only one y!

5. DOMAIN AND RANGE The set of all x values. DEFINITIONOF RANGE The set of all y values. Also called “functional values”. DEFINITIONOF DOMAIN

6. THE FUNCTION AS A “MAPPING” 1 4 7 -2 8 2 0 -3 x-values y-values y-values DOMAINRANGE Ordered Pairs: (1, 2) (4, 8) (7, - 3) (- 2, 0)

7. THE FUNCTION AS A “MAPPING” Frank Frank Sue Sue A B C Jill Jill Consider 3 students whose names are mapped to their letter grades on the last History exam: For each person in the domain, there can be only one associated letter grade in the range.

8. THE SQUARING FUNCTION -2 0 1 0 1 4 9 2 3 Each element in the domain maps to its square.

9. COUNTER-EXAMPLE: 5 1 2 3 4 Ordered Pairs: (4, 1) (4, 1) (4, 2) (4, 2) (5, 3) (5, 3) This is an example of a relation but not a function.

10. THREE WAYS TO REPRESENT A FUNCTION u NUMERICALLY - ordered pairs u SYMBOLICALLY - equation u GRAPHICALLY - picture

11. EXAMPLE Determine whether the relation represents a function: (a) {(1,4),(2,5),(3,6),(4,7)}

12. EXAMPLE Determine whether the relation represents a function: (a) {(1,4),(2,4),(3,5),(6,10)}

13. EXAMPLE Determine whether the relation represents a function: (a) {( - 3,9),(- 2,4),(0,0),(1,1),( - 3,8)}

14. EXAMPLE Determine whether the relation represents a function: (a) {( - 3,9),(- 2,4),(0,0),(1,1),( - 3,8)}

15. EVALUATING A FUNCTION AT A GIVEN X-VALUE f(x) = f(x) = f(0) = f(0) = f(2) = f(2) = f(-2) = f(-2) = f(9) = f(9) = x 2 0 4 4 81 0 0 0 0 2 4 2 4 - 2 4 9 81 9 81 x f(x) x f(x)

16. Symbolically, the squaring function can be represented as y = x 2 “FUNCTIONAL NOTATION” f(x) = x 2 Read:“f of x equals x squared”

17. EVALUATING A FUNCTION AT A GIVEN X-VALUE For f(x) = 2x 2 – 3x, find the values of the following: (a) f(3)(b) f(x) + f(3)(c) f(-x) (d) (d) - f(x)(e) f(x + 3) (f)

18. FINDING VALUES OF A FUNCTION ON A CALCULATOR DO EXAMPLE 7

19. IMPLICIT FORM OF A FUNCTION When a function is defined by an equation in x and y, we say that the function is given implicitly. If it is possible to solve the equation for y in terms of x, then we write y = f(x) and say that the function is given explicitly. See examples on Pg 127.

20. DETERMINING WHETHER AN EQUATION IS A FUNCTION Determine if x 2 + y 2 = 1 is a function. This means that for certain values of x, there are two possible outcomes for y. Thus, this is not a function!

21. Important Facts About Functions: 1.For each x in the domain of a function f, there is one and only one image f(x) in the range. For every x, there is only one y. 2.f is the symbol we use to denote the function. It is symbolic of the equation that we use to get from an x in the domain to f(x) in the range. f(x) is another name for y.

22. Important Facts About Functions: 3.If y = f(x), then x is called the independent variable or argument of f, and y is called the dependent variable or the value of f at x.

23. DOMAIN OF A FUNCTION If a function is being described symbolically and it comes with a specific domain, that domain should be expressly given. Otherwise, the domain of the function will be assumed to be the “natural domain”.

24. EXAMPLE: f(x) = x 2 Knowing the function of squaring a number, we can determine that the natural domain is all real numbers because any real number can be squared. We can also look at a graph.

25. EXAMPLE: f(x) = x 2 + 5x This is simply a modification of the squaring function. Thus, we can determine that the natural domain is all real numbers. We can also look at a graph.

26. EXAMPLE Find the domain: D: { x  }

27. EXAMPLE Find the domain: 4 – 3t 0 3t 4

28. EXAMPLE Find the domain : f(x) = x 3 + x - 1 All real numbers

29. OPERATIONS ON FUNCTIONS Notation for four basic operations on functions: Notation for four basic operations on functions: (f + g)(x) = f(x) + g(x) (f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x) (f - g)(x) = f(x) - g(x)  (f  g)(x) = f(x)  g(x) (f / g)(x) = f(x) / g(x) (f / g)(x) = f(x) / g(x)

30. OPERATIONS ON FUNCTIONS Do Example 10

31. CONCLUSION OF SECTION 1.1 CONCLUSION OF SECTION 1.1