Chapter One :

Functions, Graphs and Graphers.

The idea of a function.

The distance, in miles, is 1/5 of the time, in seconds.

15 seconds --> 3 miles

x yOrdered pairs

(x,y)Correspondenc

e

0 0 (0,0) 0 0

1 1/5 (1,1/5) 1 1/5

5 1 (5.1) 5 1

10 2 (10,2) 10 2

15 3 (15,3) 15 3

y = (1/5)x

Distance is a function of time.

A function is a correspondence between a first set, called

domain, and a second set, called the co-domain, and

only the elements which have a pre-image in the domain, make

up the range.

Notation –

inputs (members of the domain) are values of x

outputs (members of the range) are values of y

function f

input x

corresponding output f(x)

f(x) does not mean f*x

?

Vertical Line Test

If it is possible for a vertical line to cross a graph more than

once, then the graph is not a graph of a function.

Finding the domain –

The input should give real output

f(x)=3x

1

3}xR,x|{xdomain

Applications of Functions

Speed of Sound

245724575t

1087.7S(t)

t=temperature in Fahrenheit; S=speed of sound in feet/sec

boeing.com/commercial/industry_info.html

X Y

-400 469.099

-300 678.823

-200 837.599

-100 970.737

0 1087.700

100 1193.292

200 1290.197

300 1380.511

400 1464.967

500 1544.955

Temp

Sp

eed

Temp

Speed

Temp

Sp

eed

Temp

Temp

491.4t

24575t2457

24575t0

245724575t

1087.70

Anthropology

humerus

X F(x)

26 142.98

29 151.23

22 131.98

Humerus

Heig

ht

Heig

ht

y = 2.5x + 71.48

Heig

ht

Humerus

Civil Engineering

runrise

slope

For a run of hundred feet, a rise of twenty feet is permitted.

5x

f(x)

x = run f(x) = rise

Meteorology

Tem

p in

F

Month

polynomial function

0.2607392x2 + 3.620379x + 72.090909

400 ft 120t

d

Express d as a function of t

120(400)d

t

(400)(120t)d

22

22

Population

rt0eN N

N0 is the starting population N is the population after a certain time,

t , has elapsed, r is the rate of natural increase (birth rate - death rate)

and e is the constant 2.71828...

Microbiologists

Time Cells

0 1

1 2

2 4

3 8

4 16

5 32

6 64

7 128

8 256

9 512

10 1024

11 2048

Time (sec)

Cells

x2f(x)

A cardioid is a curve that can be produced as the path (or locus) of a point on the circumference of a circle as that circle rolls around another fixed circle with the same radius. wiki

Mathworld.wolfram.com

Cos = redExp = black Mathworld.wolfram.com

Hoffman-Singleton graph

Mathworld.wolfram.com

Variation

Direct variation

Inverse variation