DC Chap1A Functions

FUNCTIONSChapter 1Section 1

Prepared by Maria Cristina R. Tabuloc

1

BASIC CONCEPTS OF

FUNCTIONSSection 12

INTRODUCTIONFunctions have several uses and are

commonly encountered everyday, like finding the profit in producing products, the revenue in selling commodity, the gasoline consumption, the speed of a machine, amount time needed for a project to finish, etc. Functions can also use to project the budget for the next five years, identify the movement of a storm, to make pictures move like cartoon movies, etc.

3

A function is a relation in which

each element of the domain (set of

values of x) is associated with one

and only one element of the range

(set of values of y). However, if a

single value of y corresponds to

more than one value of x, the

relation thus obtained is also

considered a function.

DEFINITION OF FUNCTIONS

4

We may think of a function as a mapping; a function maps a number or a value from set A to one and only one value of set B.

Two values in one set could map to one value, but one value must never map to two values; that would be a relation but not a function. In notation,

y = f (x ) or f : A B

DEFINITION OF FUNCTIONS

5

FUNCTIONAL NOTATION

A function is represented by the

notation

y = f(x)where

y is called the dependent variable

x is called the independent variable

and f is the symbol used to denote that

the relation is a function.6

FUNCTIONAL NOTATION

The set of the values of

independent variable (x) is called

the domain.

The set of the values of dependent

variable (y) is called the range.

Symbols used other than f are g,

h, F, G, or H.7

ILLUSTRATION: DOMAIN & RANGE

8

For example, if we write (or define) a

function as: f(x) = x2

then we say: 'f of x equals x squared'

and we have

f( 1) = 1

f(1) = 1

f(7) = 49

f(.5) = .25

This function f maps numbers to their

squares.

FUNCTIONAL NOTATION

9

A function is a like a machine where it accepts input and processes this input to produce a product or an output.

x y

The input is the value of the independent variable x, the process involves the operations included in the function and the output is the value of the dependent variable y.

input process output

FUNCTIONAL NOTATION

10

A function may contain more than one

independent variable. Consider the relation z = 150x + 200y where x and y are the independent variables and z is the dependent variable. This function can be

written in general notation as

z = f(x, y)

If there are n independent variables, then fis written in general form as

y = f(x1, x2, x3, ).

FUNCTIONAL NOTATION

11

GRAPHS OF FUNCTIONSGraphically, a relation can be determined as to

whether it is a function or not by using the

vertical line test. When several vertical lines

are drawn through the graph and none of these

lines intersect the graph at more than one point,

then the relation obtained is a function.

y

x

y

x

y

x

y

x12

EVALUATIONS OF

FUNCTIONSSection 213

EVALUATION OF FUNCTIONS

To evaluate a function means to

solve for the dependent variable

when the independent variables are

given.

The substitution property is best to

apply in this process.

14

EXAMPLE 1

Solve the following.

1. If f(x) = x3 + 2x2 + 5, then find

a. f(2)

b. f(3/2)

c. f(1/a)

d. f(2b + 1)

15

EXAMPLE 1Solve the following.

1. If f(x) = x3 + 2x2 + 5, then find

Solution:

a) f(2) = (2)3 + 2(2)2 + 5 = 8 + 2(4) + 5 = 5 .

b) f(3/2) = (3/2)

3 + 2(3/2)2 + 5

= 27/8 + 2(9/4) + 5 =

a) f(2b + 1) = (2b + 1)3 + 2(2b + 1)2 + 5

= 8b3 + 12b2 + 6b + 1 + 2(4b2 + 4b + 1) + 5

= 8b3 + 12b2 + 6b + 1 + 8b2 + 8b + 2 + 5

= 8b3 + 20b2 + 14b + 8

8

49

552 23 2121311

aaaaaf 3

3521

a

aa

16

PROBLEMS FOR DISCUSSION 1

See the problems

Check your answers

17

OPERATIONS ON

FUNCTIONSSection 318

OPERATIONS ON FUNCTIONS

Operations on functions are similar to that

of the real numbers. The following are the

properties on functions.

Consider the functions f and g.

Sum of Functions: (f + g)(x) = f(x) + g(x)

Difference of functions: (f g)(x) = f(x) g(x)

Product of functions: (f g)(x) = f(x) g(x)

Quotient of functions: , g(x) 0

Composite functions: (f g)(x) = f( g(x))

)(

)(

xg

xf

19


Determine the indicated operations.

Given

, g(x) = x 2 4.

a) (f + g)(x) d) (g f)(x)

b) (f g)(x) e) f 2(x) = (ff)(x)

c) (f / g)(x)

5

2)(

x

xxf

Solution

20


Let f(x) = 5x and g(x) = x 2 + 3. Evaluate the following

a) (f + g)(3) e) (f g)(1)

b) (f g)(0) f) (g f)( )

c) (g f)(3/2) g) (f g)(2)

d) (f / g)(3) h) (g f)(4)

Solution21

ODD & EVEN

FUNCTIONS22

ODD AND EVEN FUNCTIONS

An even function is one whose graph is

symmetric with respect to the y-axis, and an

odd function is one whose graph is symmetric

with respect to the origin.

Formal definition:

A function f is an even function if for every

x in the domain x of f, f(x) = f(x)

A function f is an odd function if for every x

in the domain x of f, f(x) = f(x)

Graph of Odd

functionGraph of Odd

functionGraph of Even

function

Graph of Even

function

23

GRAPHS OF EVEN FUNCTIONS

24

GRAPHS OF ODD FUNCTIONS

25

xxxxf 252)( 35

4

43)(

2

3

x

xxxf

TYPES OF FUNCTIONS

& THEIR GRAPHSSection 226

TYPES OF FUNCTIONS

Polynomial Functions

Linear Functions

Quadratic Functions

Cubic Functions

Rational Functions

Split or Piecewise Functions

Absolute Value Functions

Greatest Integer Functions 27

LINEAR EQUATIONS

General Form:

Standard Forms:

Two-point Form:

Point-Slope Form:

Where

Slope-Intercept Form

Two-Intercept Form28

0 CbyAx

)()( 112

121 xx

xx

yyyy

)()( 11 xxmyy

The graph of a linear function is a

straight line and it can be written as

where m is the slope and b is the y-

intercept which are both constants.

The domain is the set of real numbers

The range is also the set of real

numbers

I. LINEAR FUNCTIONS

bmxxf )(

29

Example 2.1

Graph the function f(x) = 3x + 2

To graph, solve for two points of the

function by intercept method (two points

are enough to graph a straight line) then

plot the points on a rectangular plane.

Another way is to use the slope and the

y-intercept, m = 3 and b = 2 .

I. LINEAR FUNCTIONS

x 3 0y 7 2

30

Graph of linear function f(x) = 3x + 2

Domain = { x| x is a set of all real numbers}

Range = { y| y is a set of all real numbers}

I. LINEAR FUNCTIONS

-12

-10

-8

-6

-4

-2

0

2

4

6

-5 -4 -3 -2 -1 0 1 2

y

x

31

Definition: A quadratic function is

represented by

where a, b and c are real constants and a 0.

The domain of a quadratic function is

the set of real numbers.

II. QUADRATIC FUNCTIONS

cbxaxxf 2)(

32

The domain of a quadratic function is the

set of real numbers.

The graph of a quadratic function is a

parabola that opens upward or downward

whose vertex is

which is the maximum or minimum point


33

a

bac

a

bV

4

4,

2

2

Example 2.2: Graph the function

f(x) = 2x2 5

where 5 x 5

To graph, use the standard form of

quadratic equation then plot the vertex,

Assume some values to the right and

left of the x coordinate of the vertex,

then solve for y from each value of x to

get some points.


34

Graph of quadratic function f(x) = 2x2 5,

5 x 5


-10

-5

0

5

10

15

20

25

30

35

40

45

50

-6 -4 -2 0 2 4 6

x

y

Domain = {x|5 x 5}= [ 5, 5]

Range = {y|5 y 45}= [5, 45]

35

III. RATIONAL FUNCTIONS

Definition: A rational function is

expressed as

where p(x) and q(x) are polynomials and q(x) 0

The domain is a set of real numbers

except for values that will make q(x)

equal to zero.

)(

)()(

xq

xpxf

36

III. RATIONAL FUNCTIONS

Example 2.3: Graph the function

To graph, take several points, tabulate

the values of x and F(x).

2

6)(

+x

xxxF

2

x -3 -2 -1 0 1 2 3F(x) -6 undefined 4 3 2 1 0

37

III. RATIONAL FUNCTIONSGraph of the rational function

2

6)(

+x

xxxF

2

Notice that the graph

does not exist when

x = 2, since the function F(x) is undefined when x

= 2. This is shown by a small circle and it is

called jump of a function.

In case a jump exists, the

function is said to be

discontinuous.

Domain = {x|x R \ 2}

Range = {y| y R}38

III. RATIONAL FUNCTIONSGraph of the rational function

2

6)(

+x

xxxF

2

Notice that the graph

does not exist when

x = 2, since the function F(x) is undefined when x

= 2. This is shown by a small circle and it is

called jump of a function.

In case a jump exists, the

function is said to be

discontinuous.

Domain = {x|x R \ 2}

Range = {y| y R}39

Definition: A split or piecewise-defined

function is a function whose definition is

given differently on disjoint subsets of its

domain.

Example 2.4 Graph the function

0,22

0,2)(

2

2

xx

xxxf

x < 0 f(x) = 2x 2 x 0 f(x) = 2x 2 + 2

0.01 0.0002 0 0 2

1 2 1 42 8 2 103 18 3 20

IV. SPLIT OR PIECEWISE FUNCTION

40

Graph of

GRAPH OF PIECEWISE FUNCTION

0,22

0,2)(

2

2

xx

xxxf

Domain = {x | x R}41

Range = {y | y = (, 0) [2, )}

Definition: An absolute value

function is a function whose values

are denoted by |x| and is defined as

The domain of absolute value

function f(x) =|x| is a set of real

numbers and the range is the set of

positive real numbers.

V. ABSOLUTE VALUE FUNCTION

0

0)(

xifx

xifxxxf

42

Example 2.5: Graph f(x) = 5 + |x 3|

V. ABSOLUTE VALUE FUNCTION

0335

033535)(

xifx

xifxxxf

43

x 0 1 2 3 4 5 6

y 8 7 6 5 6 7 8

Graph of

EXAMPLES ABSOLUTE VALUE FUNCTION

0335

033535)(

xifx

xifxxxf

Domain = {x|x R}

Range = {y|y 5}

44

PROBLEMS FOR DISCUSSION 1 Graph each of the following. Determine the domain

and use the graph to find the range.

1) f(x) = x3 x2 6x

0,12

0,62)()2

xx

xxxG

2

1)()3

x

xxg

0,1

0,)()4

2

2

xx

xxxh

xxxF 13)()5

Table of valuesGraph

Graph (Exer4)

Graph(Exer5)

Graph (Exer3)

Graph(Exer2)

45

26)()6

2

x

xxxf

4

1)()7

2

xxH

Graph

(ex2)

Graph

(ex3)

46

PROBLEMS FOR DISCUSSION 1

4)()8 2 xxgGraph

(ex 4)

Definition: The greatest integer function

is represented by

where , n is an

integer.

In particular,

VI. GREATEST INTEGER FUNCTION

xxf )( 1 nxnifnx

15.1 20.2 28.2 22

57.4 07.0

44.3

32.2

21.1 31.3 47

The domain of the greatest integer

function is the set of all real

numbers

Its range consists of all the integers


Graph

(Ex 9)

48


Example 2.6: Graph

If 3 x < 2.5, h(x) = 7, If 0 x < 0.5, h(x) = 1

If 2.5 x < 2, h(x) = 6, If 0.5 x < 1, h(x) = 0

If 2 x < 1.5, h(x) = 5, If 1 x < 1.5, h(x) = 1

If 1.5 x < 1, h(x) = 4, If 1.5 x < 2, h(x) = 2

If 1 x < 0.5, h(x) = 3, If 2 x < 2.5, h(x) = 3

If 0.5 x < 0, h(x) = 2, If 2.5 x < 3, h(x) = 4

12)( xxh

Graph49

VI. GREATEST INTEGER FUNCTION:Example 2.7: Graph

If 3 x < 2, h(x) = 5 x

If 2 x < 1, h(x) = 4 x

If 1 x < 0, h(x) = 3 x

If 0 x < 1, h(x) = 2 x

If 1 x < 2, h(x) = 1 x

If 2 x < 3, h(x) = x

Graph (Ex4)

50

xxxh 2)(

EXERCISEGraph the following

222)( xxxf

31)( xxxg

Solution

Graph (Exer7)

Graph(Exer6)

51

31)( xxxxh Graph (Exer8)

5,3011

51,4

3

1,1

)(

2

2

2

xxx

xx

x

xxx

xFGraph (Exer9)

Solution

READY FOR QUIZ 1

Topic: Functions

Basic Definitions

Evaluation of Functions

Operations on Functions

Graphs of Functions & their Domain &

Range

52Ready extra short bond papers

DOMAIN & RANGE

OF FUNCTIONSChapter 1Section 3

53

DOMAIN & RANGE

Note: To find the range, determine the inverse of the function, and then examine the values of f depending on the x values.

The domain of the sum of two functions is the intersection of their domains.

54

EXAMPLE

Determine the domain and range

of each of each given function.

1) f(x) = x2 for all real number x

dom(f) = (, ) or {x| x R }

As x runs through the real numbers,

x2 runs through all the nonnegative

numbers, thus

range(f) = [0, ) 55

EXAMPLE

The variable x can take on any value, thus, dom(h) = (, )

As x approaches , h(x) also approaches ; in symbol x , h(x)

The same way, as x , h(x) . Thus, range(h) = (, ).

1)()2 3 xxh

56

EXAMPLE

For F(x) to be defined, the denominator x + 3 0 or x 3, thus, dom(F) = (, 3) ( 3,+) or

dom = {x| x R\ 3} the function has horizontal asymptote y = 1, thus,

range(F) = (, 1) ( 1,+)

3)()3

x

xxF

57

EXAMPLE

For g(x) to be defined, x 4 must be positive or zero, that is, x 4 0 or x 4 dom(g) = [4, +)

For x = 4, g(x) = 0; as x +, g(x) +

range(g) = [0, +)

4)()4 xxg

58

EXAMPLE

For H(x) to be defined, the denominator must not be zero and the radicand must be positive. Recall the solutions of nonlinear inequalities; determine the critical numbers then use the table of signs, we have

(3 x)(3 + x) > 0 the critical values are 3 and 3

29

1)()5

xxH

59

DOMAIN OF EX 5

29

1)(

xxH

(,3) (3, 3) (3, +)Assumed x 4 0 4

3 x + +

3 + x + +

(3 x)(3 + x) +

The positive product corresponds to the interval (3, 3); the function also have vertical asymptotes, x = 3 and x = 3 thus,

dom(H) = (3, 3)

60

The table of signs

RANGE OF EX 5

As x 3, H(x) +; as x 3,

H(x) +

For x = 0, H(x) = 1/3 which is the

lowest point of the curve. Thus,

range(H) = [1/3, )

61

29

1)(

xxH

GRAPH OF EX 5

62

EXAMPLE 6

From the given values of x, it is obvious that the domain is a set all real numbers;

dom(f) = (, )The range of a piecewise function is the

union of the ranges of each piece of function

range(f) = (-, 1)[1, ) = [ , +)

01

021)()6

2 xifx

xifxxf

63

EXAMPLE 6

From the given values of x, it is obvious that the domain is a set all real numbers;

dom(f) = (, )The range of a piecewise function is the

union of the ranges of each piece of function

range(f) = (1, ) [1, ) = [1, +)

01

021)()6

2 xifx

xifxxf

64

GRAPH OF EXAMPLE 6

65

EXAMPLE 7

dom(z) = [3, +)

range(z) = { z | z = 2, (1, +)}

or [2, 2] (1, +)

12

12

1332

)()7

xifx

xifx

xifx

xz

66

EXAMPLE 8

dom(G) = (, +)

range(G) = {integers}

]]32[[)()8 xxG

67

EXAMPLE 8

dom(y) = (, +)

range(y) = (2, 1]

xxxy 2]]12[[)()9

68

EXAMPLE 10

dom(G) = (, +)

range(G) = (, +)

12]]3[[)()10 xxxv

69

DOMAIN OF

COMBINATION

FUNCTIONSChapter 1

Section 4

70

COMBINATION OF FUNCTIONS

The domain of the sum of

two functions is the

intersection of their

domains.

71

Example 1.4.1: Given

Find: a) f + g b) f g c) f g

d) f/g e) f o gDetermine the domain of the following combinations of functions

4

1)(

x

xf 3)( xxg

72

SOLUTION TO EXAMPLE 1.4.1

a)

b)

34

1)()())((

x

xxgxfxgf

,44,3

,3),4(4,)( gfdom

34

1)()())((

x

xxgxfxgf

,44,3

,3),4(4,)( gfdom73

SOLUTION

c)

d)

34

1)()())((

x

xxgxfxgf

,44,3

,3),4(4,)( gfdom

341

34

1

)(

)())(/(

xxx

xxg

xfxgf

,44,3

,3),4(4,)/( gfdom74

SOLUTION

e)43

1))(())((

xxgfxgf

,19)19,3()( gfdom

75

DOMAIN & RANGE OF

PIECEWISE FUNCTIONChapter 2

Section 5

76

COMBINATION OF PIECEWISE

FUNCTIONS

Example 1.4.2: Given f and g

Find domain and range of the ff.

a) f + g

b) f g

c) f g

d) f/g

e) f o g 77

052

01

043

)(

xifx

xifx

xifx

xf

03

035

02

)(

xif

xifx

xifx

xg

SOLUTIONS TO EXAMPLE 1.4.2

052

01

043

)(

xifx

xifx

xifx

xf

03

035

02

)(

xif

xifx

xifx

xg

03)52(

0)35()1(

0)2()43(

)()()

xifx

xifxx

xifxx

xgxfa

5,055

}4|{024

2,022

)()(

rangexifx

yyrangexifx

rangexifx

xgxf 78

79

5,055

}4|{024

2,022

)()(

rangexifx

yyrangexifx

rangexifx

xgxf

5,)( gfrange

y4 2 1 0 1 2 4 5

80

03)52(

0)35()1(

0)2()43(

)()()

xifx

xifxx

xifxx

xgxfb

,1015

}6|{064

6,064

)()(

rangexifx

yyrangexifx

rangexifx

xgxf

,1015

6,064)()(

rangexifx

rangexifxxgxf

,16,)( gfrange

81

03)52(

0)35()1(

0)2()43(

)()()

xifx

xifxx

xifxx

xgxfc

6,03)52(

}5|{0)35()1(

8,0)2()43(

)()(

rangexifx

yyrangexifxx

rangexifxx

xgxf

6,)( gfrange

EXAMPLE 1.4.3

22

201

04

)(2

1

xifx

xifx

xif

xf

x

22

1

224

2

)( 2

xifx

xifx

xifx

xg

82

FUNCTIONS AS

MATHEMATICAL MODELSSection 383

WORD PROBLEMS INVOLVING

FUNCTIONS AS MATHEMATICAL MODELSSteps

Read the problem carefully, draw

figures or graphs if necessary that

will represent the problem situation.

Indicate in the figure/graph created

the known values and unknown

values. Define from these values the

dependent and independent

variables. Use appropriate symbol

(or letter) for these values.

84

WORD PROBLEMS INVOLVING FUNCTIONS

AS MATHEMATICAL MODELS

Steps (continuation)

Create the equation that defines the

function.

Solve for the unknown variable(s)

using the function formulated.

Write a conclusion consisting of

one or more sentences that

answers the questions of the

problem. 85

EXAMPLE 3.1: SOLVE THE FOLLOWING PROBLEMS

The price (y) of a jacket is equal to

p 2 500 increased by p 80 times the

quantity demanded (x).

Write a mathematical model

between the price and the quantity

demanded.

Find the price if there are 200, 220,

and 225 jackets produced.86


(From TC7 Leithold, Exer 1.3) The daily

payroll for a work crew is directly

proportional to the number of

workers, and a crew of 12 workers

earns a payroll of $810

Find a mathematical model expressing

the daily payroll as the function of the

number of workers.

What is the daily payroll for a crew of 15

workers? 87

The Kriscialou Transport charges p

10 per kilometer plus an additional

of p 5 per kilometer exceeding 100

km of the distance (x) traveled in

using a van. No additional charges

for 100 km or less.

Write the mathematical relation as

stated above. Let A be the amount

charged.


88

Graph (Example5 3)

Make a graph of the function and

determine the domain and range.

How much are the charges if the

distance traveled is 150 km, 200

km, 225 km?

How much will be charged for the

90 km distance traveled?


89

Graph (Example5 3)

The amount (V) of cement needed in

constructing a firewall is equal to the area

of the wall (A) times its thickness (t).

Write the relation if A = hl (where h is the

height and l is the length of the wall)

From (a), what is the amount of cement

needed if the height is 10 m, the length is 5

m and the thickness is 12 cm? (1m=100cm)

Write the relation if A = x2 and t = 4 in.

From (c), what is the amount of cement

if x = 6 m.

EXAMPLE 3.4: SOLVE THE FOLLOWING PROBLEM

90

(From TC7 by Leithold) In a forest a

predator feeds on prey, and on the first

fifteen (15) weeks since the end of the

hunting season, the predator

population is a function f(x), x is the

number of prey in the forest, which in

turn is a function g(t), t is the number of

weeks that have elapsed since the end

of the hunting season. If


;502)( 2481 xxxf 524)( ttg 91


the predator population as a function of

the number of weeks since the end of

the hunting season

Find the predator population 11 weeks

after the close of the hunting season.


502)( 2481 xxxf 524)( ttg

92

The amount of antibacterial solution to be

used for treating fungal infection is equal to

3cc per cm2 of the area infected. If the area

infected is more than 1 cm2, an additional of

1cc of the solution is needed and for area

less than 1 cm2, 1 cc of the solution must

be subtracted.

Write the mathematical relation stated in the

problem if x is the area infected and S is the

amount of the solution

How much solution is needed for the infected

area of 0.5cm2, 1cm2, 0.75cm2, 1.75cm2, 2.5cm2?


93

(From TC7 by Leithold) A one-story building having a rectangular floor space of 13, 200 ft2 is to be constructed where a walkway 22 ft wide is required in the front and back and a walkway of 15 ft on each side.

Find a mathematical model expressing the total area of the lot on which the building and walkways will be located as a function of the length of the front and the back of the building.

What is the domain of your function?


94

(From TC7 by Leithold) A manufacturer of

open tin boxes wishes to use the pieces

of tin with dimensions 8 in. by 15 in. by

cutting equal squares from the four

corners and turning up the sides.


the volume of the box as the function

of the length of the sides of the square

cut out.

What is the domain of your function in

the first part?

EXAMPLE 3.8: SOLVE THE FOLLOWING PROBLEM

95

READY FOR QUIZ 2

TOPICS: Functions

Domain & Range of Functions

Combination of functions

Piecewise Functions

Combination of Piecewise Function

Functions as Mathematical Models

96

Bring extra bond paper/long pad. Use only blue or black ink pen during exam. Do not use pencil or friction pen.

Documents

DC Chap1A Functions