The Function Concept

1. The Function Concept • DEFINITION: A function consists of two nonempty sets X and Y and a rule f that associates each element x in X with one and only one element y in Y. Read “The function f from X into Y” and symbolized by f : X  Y.

2. The function f from X into Y X f Y F “maps” X into Y

3. Some examples: • Supermarket item  price • Student  chair • College student  GPA • Worker  SSN • Car  license plate “number” • Real number x  x2

4. More examples: Are these functions??? • X  Y • Dormitory rooms Students • Rule: room  student(s) assigned • Airplane luggage Passengers • Rule: piece(s) of luggage  passenger • Nine digit numbers Workers • Rule: number  worker’s SSN • Real numbers Real numbers • Rule: x  the numbers y such that y2= x

5. Let X and Y be sets. A function ffrom X into Y is a set S of ordered pairs (x,y), x X, y Y, with the property that (x1, y1) and (x1, y2) are in S if and only if y1= y2. Another defintion:

6. Examples

7. Some Terminology & Notation Let f : X  Y. The set X (the “first” set) is called the domain of the function. The set of y’s in Y which correspond to an element x in X is called the range of the function.The range of f is, in general a subset of Y.

8. Variables: Let f : X  Y. The symbols x and y are called variables. In particular, a symbol such as x, representing an arbitrary element in the domain is called an independent variable. A symbol such as y, representing an element in the range corresponding to an element x in the domain is called a dependent variable.

9. Function notation: Let f : X  Y. Pick an element x in X and apply the rule f. This produces a unique element in Y. The symbol f(x) is used to denote that element. f(x) is read “f of x” or “the value of f at x” or “the image of x under f .

10. Another picture X Y f x  f(x) 

11. More pictures Y X f f(X) “Black box” x f f(x)

12. One-to-one functions: Let f : X  Y. f is a one-to-one function if it takes distinct elements in the domain to distinct elements in the range. That is: f is one-to-one if x1 x2 implies f(x1) f(x2). Notation: f is 1 – 1.

13. Examples: Which of these function is 1 – 1? • Supermarket item  price • Student  GPA • Car  license plate “number” • f(x) = 2x + 3

14. Inverse functions Suppose f : XY is 1 – 1. Then there is a function g: f(X)X such that g(f(x)) = x for all x  X. g is called the inverse of f and is denoted by f -1 f Y X f(X) g

15. Functions in Mathematics • From Geometry and Measurement: • Length function: x is a line segment, • l(x) = the length of x. • Area functions: x is a rectangle, • A(x) = the area of x. • 3. Volume functions: x is a sphere, • V(x) = the volume of x. • From Probability & Statistics: • E is a subset (event) in a sample space S, • P(E) = the probability that E “occurs”.

16. Functions in “Algebra” Let f : X  Y where X is a given set of real numbers and Y is the set of all real numbers. “fis a real-valued function of a real variable” Note: The domain X may or may not be the set of all real numbers. Examples:

17. Graph of a function Let f : X  Y. The graph of f is the set of points (x, f(x)) plotted in the coordinate plane: Graph of f = {(x, f(x)) | x X }. The graph of f is a “geometric” object – a “picture” of the function.

18. Examples:

19. Functions defined on the positive integers: Sequences A function f whose domain is the set of positive integers is called a sequence. The values are called the terms of the sequence; f(1) is the 1st term, f(2) is the 2nd term, and so on

20. Subscript notation It is customary to use subscript notation rather than functional notation: and to denote the sequence by an

21. Examples

22. Recursion formulas A recursion formula or recurrence relation gives ak+1in terms of one or more of the terms amthat precede ak+1. Examples: Find the first four terms and the nth term for the sequence specified by

23. Solutions

24. More examples • List the first six terms of the sequence whose nth term an is the nth prime number. Give a “formula” for an. • (4) The first four terms of the • sequence anare: • What is the 5thterm?

25. Answers • 2, 3, 5, 7, 11, 13; an = ?????? • (2)

26. Limits of sequences Given a sequence an. What is the behavior of anfor very large n ? That is, as n what can you say about an ? Examples:

27. Answers • 1 (2) 0 • (3) No limit (4) No limit

28. Two special sequences • Arithmetic sequences: A sequence is an arithmetic sequence (arithmetic progression) if successive terms differ by a constant d, called the common difference. That is anis an arithmetic sequence if

29. Examples • Answers: • Yes • No • Yes, assuming the pattern goes on as • indicated

30. What is the 12th term of the arithmetic sequence whose first three terms are: • 1, 5, 9?

31. Solving the recursion formula

32. Geometric sequences A geometric sequence is a sequence in which the ratio of successive terms is a nonzero constant r. That is, The number r is called the common ratio.

33. Examples • The sequence 8, 4, 2, 1, …. is a geometric sequence. Find the common ratio and give the 5th term. • The sequence • is a geometric sequence, find the common ratio and give the 6th term. • (3) angeometric sequence with common ratio r. Give a formula for an.

34. Answers:

35. Function defined on intervals Let f : X Y where X is an interval or a union of intervals and Y is the set of real numbers. The graph of f is the set of all points (x,f(x)) in the coordinate plane. The graph of f is the graph of the equation y=f (x).

36. Examples f(x) = 2x + 1 f (x) = x2 + 1

37. The Elementary Functions • The constant functions: The graph of f is a horizontal line c units above or below the x-axis depending on the sign of c. f (x) = 2

38. (2) The identity function and linear functions (a) The function f (x) = x is called the identity function. The graph is

40. NONLINEAR FUNCTIONS a > 0 a < 0

41. a > 0 a < 0

42. (5) Polynomial Functions

44. (6) Rational functions

45. Some graphs

46. The Elementary Functions • Algebraic functions: sums, differences, products, quotients and roots of rational functions. • The trigonometric functions. • Exponential functions. • Logarithm functions.