SECTION 1.1SECTION 1.1

FUNCTIONSFUNCTIONS

DEFINITION OF A RELATION

DEFINITION OF A RELATION

A relation is a correspondence A relation is a correspondence between two sets. If x and y between two sets. If x and y are two elements in these sets are two elements in these sets and if a relation exists between and if a relation exists between x and y, we say that x x and y, we say that x corresponds to y or that y corresponds to y or that y depends upon x. The depends upon x. The correspondence can be written correspondence can be written as an ordered pair (x,y).as an ordered pair (x,y).

DEFINITION OF A RELATION

DEFINITION OF A RELATION

Thus, a relation is simply a set Thus, a relation is simply a set of ordered pairs or a table of ordered pairs or a table which relates x and y values.which relates x and y values.

DEFINITION OF FUNCTION

DEFINITION OF FUNCTION

Let X and Y be two nonempty Let X and Y be two nonempty sets of real numbers. A sets of real numbers. A functionfunction from X into Y is a rule from X into Y is a rule or a correspondence that or a correspondence that associates with each element of associates with each element of X a X a uniqueunique element of Y. This is element of Y. This is a special type of relation.a special type of relation.For every x, there is only For every x, there is only

one y!one y!

DOMAIN AND RANGEDOMAIN AND RANGE

The set of all x The set of all x values.values.

DEFINITIODEFINITIONNOFOF

RANGERANGE

The set of all y The set of all y values.values.

Also called Also called “functional “functional values”.values”.

DEFINITIODEFINITIONNOFOF

DOMAINDOMAIN

THE FUNCTION AS A “MAPPING”

THE FUNCTION AS A “MAPPING”

11

44

77

--22

88

22

00--33

x-x-valuesvalues

y-y-valuesvalues

DOMAIDOMAINN

RANGERANGE

Ordered Ordered Pairs:Pairs:(1 , (1 ,

2)2)(4 , (4 , 8)8)(7, - (7, - 3)3)(- 2, (- 2, 0)0)

THE FUNCTION AS A “MAPPING”

THE FUNCTION AS A “MAPPING”

FrankFrank SueSue

AA

BB

CC

JillJill

Consider 3 students whose Consider 3 students whose names are mapped to their letter names are mapped to their letter grades on the last History exam:grades on the last History exam:

For each person For each person in the domain, in the domain, there can be there can be only one only one associated associated letter grade in letter grade in the range.the range.

THE SQUARING FUNCTION

THE SQUARING FUNCTION

--22--1100

11

00 11 44 99

22 33

Each Each element in element in the domain the domain maps to its maps to its square.square.

COUNTER-EXAMPLE:COUNTER-EXAMPLE:

55

11 22 33

44Ordered Pairs:Ordered Pairs:

(4, 1)(4, 1)

(4, 2)(4, 2)

(5, 3)(5, 3)

This is an example of a This is an example of a relation but not a function.relation but not a function.

THREE WAYS TO REPRESENT A FUNCTION

THREE WAYS TO REPRESENT A FUNCTION

NUMERICALLY - ordered NUMERICALLY - ordered pairspairs

SYMBOLICALLY - equationSYMBOLICALLY - equation

GRAPHICALLY - pictureGRAPHICALLY - picture

EXAMPLEEXAMPLE

Determine whether the Determine whether the relation represents a function:relation represents a function:

(a) {(1,4),(2,5),(3,6),(4,7)}(a) {(1,4),(2,5),(3,6),(4,7)}

EXAMPLEEXAMPLE

Determine whether the Determine whether the relation represents a function:relation represents a function:

(a) {(1,4),(2,4),(3,5),(6,10)}(a) {(1,4),(2,4),(3,5),(6,10)}

EXAMPLEEXAMPLE

Determine whether the Determine whether the relation represents a function:relation represents a function:

(a) {( - 3,9),(- 2,4),(0,0),(1,1),(a) {( - 3,9),(- 2,4),(0,0),(1,1),( - 3,8)}( - 3,8)}

EXAMPLEEXAMPLE

Determine whether the Determine whether the relation represents a function:relation represents a function:

(a) {((a) {( - 3- 3,9),(- 2,4),(0,0),(1,1),,9),(- 2,4),(0,0),(1,1),(( - 3- 3,8)},8)}

EVALUATING A FUNCTION AT A GIVEN X-VALUE

EVALUATING A FUNCTION AT A GIVEN X-VALUE

f(x) f(x) = =

f(0) f(0) = =

f(2) f(2) = =

f(-2) f(-2) ==

f(9) f(9) = =

x x 22

00

44

44

8181

0 00 0

2 42 4

- 2 4- 2 4

9 819 81

xx f(x) f(x)

Symbolically, the squaring Symbolically, the squaring function can be represented asfunction can be represented as

y = xy = x 2 2

““FUNCTIONAL NOTATION”FUNCTIONAL NOTATION”

f(x) = xf(x) = x 2 2

Read:Read: “f of x equals x “f of x equals x squared”squared”

EVALUATING A FUNCTION AT A GIVEN

X-VALUE

EVALUATING A FUNCTION AT A GIVEN

X-VALUEFor f(x) = 2x2 – 3x, find

the values of the following:

(a) f(3) (b) f(x) + f(3)(c) f(-x)

(d) - f(x) (e) f(x + 3)

(f)

hf(x)h)f(x

FINDING VALUES OF A FUNCTION ON A

CALCULATOR

FINDING VALUES OF A FUNCTION ON A

CALCULATOR

DO EXAMPLE 7

IMPLICIT FORM OF A FUNCTION

IMPLICIT FORM OF A FUNCTION

When a function is defined by an equation in x and y, we say that the function is given implicitly. If it is possible to solve the equation for y in terms of x, then we write y = f(x) and say that the function is given explicitly. See examples on Pg 127.

DETERMINING WHETHER AN EQUATION IS A

FUNCTION

DETERMINING WHETHER AN EQUATION IS A

FUNCTION

Determine if x2 + y2 = 1 is a function.

2x1y

This means that for certain values This means that for certain values of x, there are two possible of x, there are two possible outcomes for y.outcomes for y.Thus, this is not a function!Thus, this is not a function!

Important Facts About Functions:

Important Facts About Functions:

1.1. For each For each xx in the domain of a in the domain of a function function ff, there is one and only , there is one and only one image one image f(x)f(x) in the range. For in the range. For every every xx, there is only , there is only one one yy..

2.2. ff is the symbol we use to is the symbol we use to denote the denote the function. It is symbolic function. It is symbolic of the of the equation that we use to equation that we use to get from an get from an xx in the domain to in the domain to f(x)f(x) in the range. in the range. f(x)f(x) is another is another name for name for yy..

Important Facts About Functions:Important Facts About Functions:

3.3. If y = If y = f(x)f(x) , then , then xx is called the is called the independent variable or argument independent variable or argument of of ff, and , and yy is called the dependent is called the dependent

variable or the value of variable or the value of ff at at xx..

DOMAIN OF A FUNCTION

DOMAIN OF A FUNCTION

If a function is being If a function is being described symbolically and it described symbolically and it comes with a specific domain, comes with a specific domain, that domain should be that domain should be expressly given.expressly given.

Otherwise, the domain of the Otherwise, the domain of the function will be assumed to function will be assumed to be the “natural domain”.be the “natural domain”.

EXAMPLE: f(x) = x 2EXAMPLE: f(x) = x 2

Knowing the function of Knowing the function of squaring a number, we can squaring a number, we can determine that the natural determine that the natural domain is all real numbers domain is all real numbers because any real number can because any real number can be squared.be squared.

We can also look at a graphWe can also look at a graph..

EXAMPLE: f(x) = x 2 + 5x

EXAMPLE: f(x) = x 2 + 5x

This is simply a modification This is simply a modification of the squaring function. of the squaring function. Thus, we can determine that Thus, we can determine that the natural domain is all real the natural domain is all real numbers.numbers.

We can also look at a graphWe can also look at a graph..

EXAMPLEEXAMPLE

Find the Find the domain:domain: 42x

3x = f(x)

D: { xD: { x } }

2x

2x

EXAMPLEEXAMPLE

Find the Find the domain:domain:

3t-4 = f(t)

4 – 3t 0 4 – 3t 0

3t 43t 4

34

t

EXAMPLEEXAMPLE

Find the domain Find the domain ::

f(x) = xf(x) = x33 + x - + x - 11

All real numbersAll real numbers

OPERATIONS ON FUNCTIONS

OPERATIONS ON FUNCTIONS

Notation for four basic Notation for four basic operations on functions:operations on functions:

(f + g)(x) = f(x) + g(x)(f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x)(f - g)(x) = f(x) - g(x) (f (f g)(x) = f(x) g)(x) = f(x) g(x)g(x) (f / g)(x) = f(x) / g(x)(f / g)(x) = f(x) / g(x)

OPERATIONS ON FUNCTIONS

OPERATIONS ON FUNCTIONS

Do Example 10Do Example 10

CONCLUSION OF SECTION 1.1CONCLUSION OF SECTION 1.1