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.
The function f from X into Y F “maps” X into Y
Some examples: Supermarket item price Student chair College student GPA Worker SSN Car license plate “number” Real number x x2
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
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.
Examples
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.
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.
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 .
Another picture X Y f x f(x)
More pictures Y X f f(X) “Black box” x f f(x)
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.
Examples: Which of these function is 1 – 1? Supermarket item price Student GPA Car license plate “number” f(x) = 2x + 3
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
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”.
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:
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.
Examples:
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
Subscript notation It is customary to use subscript notation rather than functional notation: and to denote the sequence by an
Examples
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
Solutions
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?
Answers 2, 3, 5, 7, 11, 13; an = ?????? (2)
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:
Answers 1 (2) 0 (3) No limit (4) No limit
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
Examples Answers: Yes No Yes, assuming the pattern goes on as indicated
What is the 12th term of the arithmetic sequence whose first three terms are: 1, 5, 9?
Solving the recursion formula
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.
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.
Answers:
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).
Examples f(x) = 2x + 1 f (x) = x2 + 1
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
(2) The identity function and linear functions (a) The function f (x) = x is called the identity function. The graph is
NONLINEAR FUNCTIONS a > 0 a < 0
a > 0 a < 0
(5) Polynomial Functions
(6) Rational functions
Some graphs
The Elementary Functions Algebraic functions: sums, differences, products, quotients and roots of rational functions. The trigonometric functions. Exponential functions. Logarithm functions.