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
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. Another defintion:
Let X and Y be sets. A function f from 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.

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.
“f is 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 f = {(x, f(x)) | x X }.
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+1 in terms of one or more of the terms am that 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 an are:
What is the 5th term?

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

26. Limits of sequences
Given a sequence an. What is the behavior of an for 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 an is 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) an geometric 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 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.