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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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X Y f 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 x 2

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More examples: Are these functions??? X Y Dormitory roomsStudents Rule: room student(s) assigned Airplane luggage Passengers Rule: piece(s) of luggage passenger Nine digit numbersWorkers Rule: number workers SSN Real numbersReal numbers Rule: x the numbers y such that y 2 = 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 (x 1, y 1 ) and (x 1, y 2 ) are in S if and only if y 1 = y 2.

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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 ys 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 XY x f(x)f(x) f

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

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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 x 1 x 2 implies f(x 1 ) f(x 2 ). 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 : 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 X Y f(X) f g

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Functions in Mathematics From Geometry and Measurement: 1.Length function: x is a line segment, l(x) = the length of x. 2.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 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 1 st term, f(2) is the 2 nd 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 a n

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Examples

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Recursion formulas A recursion formula or recurrence relation gives a k+1 in terms of one or more of the terms a m that precede a k+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 (3) List the first six terms of the sequence whose nth term a n is the nth prime number. Give a formula for a n. (4) The first four terms of the sequence a n are: What is the 5 th term?

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Answers (1) 2, 3, 5, 7, 11, 13; a n = ?????? (2)

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

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

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Two special sequences 1.Arithmetic sequences: A sequence is an arithmetic sequence (arithmetic progression) if successive terms differ by a constant d, called the common difference. That is a n is an arithmetic sequence if

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Examples Answers: (1) Yes (2) No (3) Yes, assuming the pattern goes on as indicated

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(4)What is the 12 th 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 (1)The sequence 8, 4, 2, 1, …. is a geometric sequence. Find the common ratio and give the 5 th term. (2) The sequence is a geometric sequence, find the common ratio and give the 6 th term. (3) a n geometric sequence with common ratio r. Give a formula for a n.

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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 + 1 f (x) = x 2 + 1

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The Elementary Functions 1.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 > 0a < 0

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a > 0a < 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 (7) Algebraic functions: sums, differences, products, quotients and roots of rational functions. (8) The trigonometric functions. (9) Exponential functions. (10) Logarithm functions.

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