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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.

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**The function f from X into Y**

F “maps” X into Y

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**Some examples: Supermarket item price Student chair**

College student GPA Worker SSN Car license plate “number” Real number x x2

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**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

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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.

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Examples

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**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.

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**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.

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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 .

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Another picture X Y f x f(x)

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

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**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.

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**Examples: Which of these function is 1 – 1?**

Supermarket item price Student GPA Car license plate “number” f(x) = 2x + 3

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**Inverse functions Suppose f : XY 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

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**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”.

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**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:

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**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.

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Examples:

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**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

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

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Examples

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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

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Solutions

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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?

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

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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:

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Answers 1 (2) 0 (3) No limit (4) No limit

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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

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**Examples Answers: Yes No Yes, assuming the pattern goes on as**

indicated

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**What is the 12th term of the arithmetic sequence whose first three terms are:**

1, 5, 9?

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**Solving the recursion formula**

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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.

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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.

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Answers:

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**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).

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

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**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

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**(2) The identity function and linear functions**

(a) The function f (x) = x is called the identity function. The graph is

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NONLINEAR FUNCTIONS a > 0 a < 0

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a > 0 a < 0

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**(5) Polynomial Functions**

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(6) Rational functions

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Some graphs

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**The Elementary Functions**

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

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Real Numbers and Algebra

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