Chapter 0

Functions

§ 0.1

Functions and Their Graphs

Definition Example

Rational Number: A number that may be written as a finite or infinite repeating decimal, in other words, a number that can be written in the form m/n such that m, n are integers

Irrational Number: A number that has an infinite decimal representation whose digits form no repeating pattern

Rational & Irrational Numbers

73205.13

285714.07

2

The Number Line

A geometric representation of the real numbers is shown below.

The Number Line

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

37

2

Open & Closed Intervals

Definition Example

Open Interval: The set of numbers that lie between two given endpoints, not including the endpoints themselves

Closed Interval: The set of numbers that lie between two given endpoints, including the endpoints themselves

[-1, 4]

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

,44x

41 x

Functions

EXAMPLEEXAMPLE

If , find f (a - 2). 342 xxxf

Domain

Definition Example

Domain of a Function: The set of acceptable values for the variable x.

The domain of the function

is

x

xf

3

1

03 xx3

Graphs of Functions

Definition Example

Graph of a Function: The set of all points (x, f (x)) where x is the domain of f (x). Generally, this forms a curve in the xy-plane.

The Vertical Line Test

Definition Example

Vertical Line Test: A curve in the xy-plane is the graph of a function if and only if each vertical line cuts or touches the curve at no more than one point.

Although the red line intersects the graph no more than once (not at all in this case), there does exist a line (the yellow line) that intersects the graph more than once. Therefore, this is not the graph of a function.

Graphing Calculators

Graphing Using a Graphing Calculator

Step Display

1) Enter the expression for the function.

2) Enter the specifications for the viewing window.

3) Display the graph.

Graphs of Equations

EXAMPLEEXAMPLE

Is the point (3, 12) on the graph of the function ? 22

1

xxxf

§ 0.2

Some Important Functions

Linear Equations

Equation Example

y = mx + b(This is a linear function)

x = a(This is not the graph of a

function)

Linear Equations

Equation Example

y = b

CONTINUECONTINUEDD

Piece-Wise Functions

EXAMPLEEXAMPLE

Sketch the graph of the following function .

3for 2

3for 1

x

xxxf

Quadratic Functions

Definition Example

Quadratic Function: A function of the form

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

cbxaxxf 2

Polynomial Functions

Definition Example

Polynomial Function: A function of the form

where n is a nonnegative integer and a0, a1, ...an are given numbers.

01

1 axaxaxf nn

nn

517 23 xxxf

Rational Functions

Definition Example

Rational Function: A function expressed as the quotient of two polynomials.

15

32

4

xx

xxxg

Power Functions

Definition Example

Power Function: A function of the form

.rxxf

2.5xxf

Absolute Value Function

Definition Example

Absolute Value Function: The function defined for all numbers x by

such that |x| is understood to be x if x is positive and –x if x is negative

,xxf

xxf

212121 f

§ 0.4

Zeros of Functions – The Quadratic Formula and Factoring

Zeros of Functions

Definition Example

Zero of a Function: For a function f (x), all values of x such that f (x) = 0.

12 xxf

10 2 x1x

Quadratic Formula

Definition Example

Quadratic Formula: A formula for solving any quadratic equation of the form .

The solution is:

There is no solution if

These are the solutions/zeros of the quadratic function

02 cbxax

0232 xx

.2

42

a

acbbx

2;3;1 cba

12

21433 2 x

2

173x

.232 xxxf.042 acb

Graphs of Intersecting Functions

EXAMPLEEXAMPLE

Find the points of intersection of the pair of curves.

;9102 xxy 9xy

Factoring

EXAMPLEEXAMPLE

Factor the following quadratic polynomial.326 xx

Factoring

EXAMPLEEXAMPLE

Solve the equation for x.

2

651

xx

§ 0.5

Exponents and Power Functions

Exponents

Definition Example

bn = b*b*b…*b55553

nn bb 1

33

1

55

Exponents

Definition Example

344 34

3

555 mnn mn

m

bbb

344 34

34

3

5

1

5

1

5

15

mnn mn

mn

m

bbb

b111

Exponents

Definition Example

666666 13

3

3

2

3

1

3

2

3

1

srsr bbb

2

1

4

1

4

14

2

12

1

rr

bb

1

Exponents

Definition Example

7777

7

7 13

3

3

1

3

4

3

1

3

4

srs

r

bb

b

399999 2

1

8

4

8

5

5

48

5

5

4

rssr bb

Exponents

Definition Example

153527125

271252712533

3/13/13/1

rrr baab

1625

10

5

10 44

4

4

r

rr

b

a

b

a

Applications of Exponents

EXAMPLEEXAMPLE

Use the laws of exponents to simplify the algebraic expression.

3

3/2527

x

x

Compound Interest - Annual

Definition Example

Compound Interest Formula:

A = the compound amount (how much money you end up with)

P = the principal amount invested

i = the compound interest rate per interest period

n = the number of compounding periods

If $700 is invested, compounded annually at 8% for 8 years, this will grow to:

Therefore, the compound amount would be $1,295.65.

niPA 1

808.01700 A

808.1700A

851.1700A

651.295,1A

Compound Interest - General

Compound Interest - General

EXAMPLEEXAMPLE

(Quarterly Compound) Assume that a $500 investment earns interest compounded quarterly. Express the value of the investment after one year as a polynomial in the annual rate of interest r.