Number and Algebra lecture 11

Polynomial rings,

Functions

History Of Function Concept

• CA 200 BC Function concept has origins in Greek and Babylonian mathematics.

• Babylonian Tablets for finding squares and roots.

• Middle Ages: mathematicians expressed generalized notions of dependence between varying quantities using verbal descriptions.

• Late 16th – Early 17th Century – Galileo and Kepler study physics, notation to support this study lead to algebraic notation for function.

• Leibniz (1646 – 1716) introduces term “function” as quantity connected to a curve.

• Bernoulli(1718) interprets function as any expression made up of a variable and constants.

• Euler (1707 – 1783) regarded a function as any equation or formula.

• Clairant (1734) developed notation f(x), functions were viewed as well-behaved (smooth & continuous).

• Dirichlet (1805-1859) introduced concept of variables in a function being related as well as each x having a unique image y.

Question

• What is your definition of function?

• Which of the following are functions under Euler’s definition? Under Dirichlet’s definition?

• x2 + y2 = 25

• f(x) = 0 if x is rational

1 if x is irrational

Function

• A relation satisfying the univalence property.

• Univalence Property: x domain(f),

a unique y range(f) such that

f(x) = y.

Function Concept Table

Representation

Object

Process

AlgebraicGraphic Numeric VerbalInterpretation

Function Translation

Curve Sketching

Computing Values

Recognize Formula

Algebraic

Curve Fitting

Reading Values

Interpret Graph

Graphic

Fitting dataPlottingReadingNumeric

ModelingSketchingMeasuringVerbal

AlgebraicGraphicNumericVerbalTo

From

Function Misconceptions• Functions must have an algebraic rule.

For every value of x choose a corresponding value of y by rolling a die.

• Tables are not functions.

7 6 4 1 2 8 7 5 3 Y

9 8 7 6 5 4 3 2 1 X

• Functions can have only one rule for all domain values.

x + 1 if x 0 y = 2x + 1 if x > 0• Functions cannot be a set of disconnected

points. x if x is even y = 2x if x is odd• Any equation represents a function. x2 + y2 = 25

More Function Misconceptions

• Functions must be smooth, they cannot have corners.

y = | x |

• Functions must be continuous.

0,1

0,

11

)1)(1(

2

xx

xxy

xy

x

xxy

Function Tests

• Geometric: Vertical Line Test

Function Tests

• Algebraic: f is a function iff

x1 = x2 implies that f(x1) = f(x2).

• Function Diagram

Domain Range

Process Interpretation of Function

• A function is a dynamic process assigning each domain value a unique range value.

FunctionDomain

Range

Input x

Output f(x)

Process Interpretation Tasks

• Evaluating a function at a point– Ex: Find f(2) when f(x) = 3x - 5

• Determining Domain and Range– Ex: Determine the domain and range of the

seven basic algebraic functions

Constant Function

Ex: f(x) = 5

Domain:

Range:

Identity Function

f(x) = x

Domain:

Range:

Square Function

f(x) = x2

Domain:

Range:

Cube Function

f(x) = x3

Domain:

Range:

Square Root Function

Domain:

Range:

xxf )(

Reciprocal Function

Domain:

Range:

xxf

1)(

Absolute Value Function

Domain:

Range:

xxf )(

Object Interpretation of Function

A function is a static object or thing

Allows for:

• Trend Analysis

• Classification

• Operation

Function as Object: Trend Analysis

The graph below represents a trip from home to school. Interpret the trends.

School

Hometime

distance

Function as Object: Classification

•A function that is symmetric to the y-axis is said to be even.

•A function that is symmetric about the origin is said to be odd.

•Classify the following as even or odd:

1. x 0 2 -2 7 -7y 5 3 3 -9 -9

Classify as even or odd:

2. 3. y = x2 + 5

4. y = x5 + 3x3 - x

Function as Object: Operation

Given two functions f(x) and g(x), we can combine them to get a new function:

))(())((

)(/)())(/(

)()())((

)()())((

)()())((

xgfxgf

xgxfxgf

xgxfxgf

xgxfxgf

xgxfxgf

Inverse

• Inverse: to turn inside out, to undo

• Additive Inverse: a + (-a) = 0

• Multiplicative Inverse: a • (1/a) = 1

• Pattern: (element) * (inverse) = identity

Function Identity

Let i(x) represent the identity, then for any function f(x) we have

Ex: f(x) = 5x + 2, then

What is i(x)?

)()()( xfxixf

2)]([5))(())(( xixifxif

Function Inverse

Given identity is i(x)=x, f -1(x) is a function such that

xxff ))(( 1

What is the inverse for the function in table/numeric form?

1. x 1 2 3 4y 2 8 7 5

2. x 1 -1 3 7y 2 2 5 8

What is the inverse for the function in graphic form?

1. 2.

What is the inverse for the function f(x)=3x+5 in algebraic form?

Abstract Algebra

• In the 19th century British mathematicians took the lead in the study of algebra.

• Attention turned to many "algebras" - that is, various sorts of mathematical objects (vectors, matrices, transformations, etc.) and various operations which could be carried out upon these objects.

MORE INFO• http://www.math.niu.edu/~beachy/aaol/frames_index.html

• Thus the scope of algebra was expanded to the study of algebraic form and structure and was no longer limited to ordinary systems of numbers.

• The most significant breakthrough is perhaps the development of non-commutative algebras. These are algebras in which the operation of multiplication is not required to be commutative.

• ((a,b) + (c,d) = (a+b,c+d) ;

• (a,b) (c,d) = (ac - bd, ad + bc)).

• Gibbs (American, 1839 -1903) developed an algebra of vectors in three-dimensional space.

• Cayley (British, 1821-1895) developed an algebra of matrices (this is a non-commutative algebra).

• The concept of a group (a set of operations with a single operation which satisfies three axioms) grew out of the work of several mathematicians

• …and then came the concepts of rings and fields

Polynomial in x with coefficients in S

• Let S be a commutative ring with unity

• Indeterminate x – symbol interpretation of variable.

• A polynomial is an algebraic expression of the form

ao xo + a1x1+ a2x2 + …. + anxn

where n Z+ U {0} ai S

• Coefficients ai.

• Polynomial in x over S.

• Term of Polynomial aixi .

Francis Sowerby MacaulayBorn: 11 Feb 1862 in Witney,

EnglandDied: 9 Feb 1937 in Cambridge,

Cambridgeshire, England

• Macaulay wrote 14 papers on algebraic geometry and polynomial ideals.

• Macaulay discovered the primary decomposition of an ideal in a polynomial ring which is the analog of the decomposition of a number into a product of prime powers in 1915.

• In other words, in today's terminology, he is examining ideals in polynomial rings.

Wolfgang KrullBorn: 26 Aug 1899 in Baden-

Baden, GermanyDied: 12 April 1971 in Bonn,

Germany

• Krull's first publications were on rings and algebraic extension fields.

• He was quickly recognized as a decisive advance in Noether's programme of emancipating abstract ring theory from the theory of polynomial rings.

Question

Which of the following are polynomials?• Let S = {ai ai is an even integer}, then is

ao xo + a1x1+ a2x2 + …. + anxn

a polynomial?• Let S = Z, then is

ao xo + a1x1+ a2x2 + …. + anxn

a polynomial?5x3 – ½ x2 + 2i x + 5 where S = C

• x -2 + 2x – 5

• x1/2 + ½ x2 + 3

• ni=0 aixi

• 2 + x3 – 2x5

Polynomial Ring

• Is (S [x],+,• ) a polynomial ring?

• Is (S [x],+,• ) a commutative ring?

• Is (S [x],+,• ) a ring with unity?

Closure +

r

i

iii xbaxgf

0

)())((

Closure •

nm

i

ii

kkik xbaxgf

0 0

)())((

Commutative & Associative for + and •

Identity +

Inverse +

Identity •