Date post: | 28-Mar-2015 |
Category: |
Documents |
Upload: | desirae-merrithew |
View: | 220 times |
Download: | 1 times |
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.
X Yf
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 StudentsRule: room student(s) assigned
• Airplane luggage PassengersRule: piece(s) of luggage passenger
• Nine digit numbers WorkersRule: number worker’s SSN
• Real numbers Real numbersRule: 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
)}6,3(),4,3(),2,1(),1,2(),1,1(),0,0{()2(
)}10,3(),10,3(),5,2(),5,2(),2,1(),2,1(),1,0{()1(
S
S
Some Terminology & Notation
Let f : X Y.
The set X (the “first” set) is called the domainof 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 anindependent 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
x f(x)f
More pictures
X Y
ff(X)
“Black box”x
f(x)f
One-to-one functions:
Let f : X Y.
f is a one-to-one function if it takes distinctelements 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 functionsSuppose 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
X Y
f(X)
f
g
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”.
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 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:
1)( 2 nnf
1)( 2 xxf
1)( 2 xxf
1)( 2 xxf
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
(1) , (2) , (3) , , ( ) ,f f f f n
Subscript notationIt is customary to use subscript notation rather than functional notation:
and to denote the sequence by an
1 2 3(1) , (2) , (3) , , ( ) ,nf a f a f a f n a
Examples
1 3 4 5; first four terms: 2, , , ,
2 3 4n
na
n
2
1 1 2 3 4; first four terms: , , ,
2 5 10 17n
na
n
( 1) ; first four terms: 1, 1, 1, 1nna
2( 1) ; first four terms: 1, 4, 9, 16nna n
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
1 1(1) 3 and 2 , 1,2,3,k ka a a k
1 1(2) 1 and ( 1) , 1,2,3,k ka a k a k
Solutions
2 1
23 2
2 34 3
1
(1) 2 2 3,
2 2 2 3 2 3,
2 2 2 3 2 3
In general, 2 3nn
a a
a a
a a
a
2 1
3 2
4 3
(2) 2 2 1,
3 3 2 1,
4 4 3 2 1
In general, !n
a a
a a
a a
a n
More examples
(3) 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?
1 1 11, , ,
2 3 4
Answers(1) 2, 3, 5, 7, 11, 13; an = ??????
(2) 2
120
1
48)4)(3)(2)(1(
1 nnnn
nan
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:
2
2
1 1(1) (2)
(3) ( 1) (4) ( 1)
n n
n nn n
n na a
n n
a a n
Answers
(1) 1 (2) 0
(3) No limit (4) No limit
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 an is an arithmetic sequence if
1 for every positive integer k ka a d k
Examples
Determine whether the sequence is an arithmetic sequence
(1) 2, 5, 8, 11, ,3 1
(2) 1, 4, 9, 16,
(3) 22, 18, 14, 10, ,
n
Answers:(1) Yes(2) No(3) Yes, assuming the pattern goes on as indicated
(4)What is the 12th term of the arithmetic sequence whose first three terms are:
1, 5, 9?
1 2 3(5) The sequence , , , is an arithmetic sequence.
What is for all ?n
a a a
a n
Solving the recursion formula
454)11(1 is (4) ofsolution The (5). solves This
)1(
3
2
Thus,
. implieswhich know We
12
1
134
123
12
11
a
dnaa
dadaa
dadaa
daa
daadaa
n
kkkk
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.
kkk
k arara
a
1
1 implieswhich
Examples
(1)The sequence 8, 4, 2, 1, …. is a geometric sequence. Find the common ratio and give the 5th term.
(2) 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.
,8
5,
4
5,
2
5,5
Answers:
111
6
5
Then.Let )3(
32
5;
2
1)2(
2
1;
2
1)1(
nn raaaa
ar
ar
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
1. The constant functions: constant. a is where)( ccxf
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
.
intercept- and slope with linestraight a is
ofgraph The . called are
0,)(
form theof Functions (b)
b
ym
ffunctionslinear
mbmxxf
12)( xxf
NONLINEAR FUNCTIONS
parabola. a isfunction quadratic a
ofgraph The . called are
0 with constants are and , where
)(
form theof Functions (3)2
functionsquadratic
acba
cbxaxxf
a > 0 a < 0
like looks cubic
a ofgraph The . called are
0 with constants are and,,, where
)(
form theof Functions (4)23
functionscubic
adcba
dcxbxaxf
a > 0 a < 0
(5) Polynomial Functions
ts.coefficien theare
,, ;polynomial theof deg theis
function. polynomial a is
,0 with constants are,,,
andinteger enonnegativ a is where
,)(
form theoffunction A
10
10
011
1
aareen
aaaa
n
axaxaxaxf
nn
nn
nn
converselynot but 1
)()( :function rational a isfunction polynomialA
on so and
functions. quadratic ;2 degree of spolynomial :2
functions.linear ;1 degree of spolynomial :1
functions.
constant nonzero the;0 degree of spolynomial :0
xpxp
n
n
n
(6) Rational functions
.1
)()(function rational a isfunction polynomialA
functions. polynomial are and where
)(
)()(
form theoffunction a is A
xpxp
qp
xq
xpxr
unctionrational f
Some graphs
2
1)(
xxf
1)(
2
x
xxf
1
12)(
2
x
xxf
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.