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 15  3 (15,3) 3 15 10  2 (10,2) 2 10 5  1 (5.1) 1 5 1  1/5 (1,1/5) 1/5 1 0  0 (0,0) 0 0 Correspondence Ordered pairs (x,y) y

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

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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)=

Applications of Functions

Speed of Sound

t=temperature in Fahrenheit; S=speed of sound in feet/sec boeing.com/commercial/industry_info.html

1544.955 500 1464.967 400 1380.511 300 1290.197 200 1193.292 100 1087.700 0 970.737 -100 837.599 -200 678.823 -300 469.099 -400 Y X

Temp Speed

Temp Speed

Temp Speed

Temp

Temp

Anthropology

humerus

131.98 22 151.23 29 142.98 26 F(x) X

Humerus Height

Height

y = 2.5x + 71.48

Height Humerus

Civil Engineering

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

x = run f(x) = rise

Meteorology

Temp in F Month

polynomial function 0.2607392 x 2 + 3.620379 x + 72.090909

400 ft 120 t d Express d as a function of t

Population

N 0 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

2048 11 1024 10 512 9 256 8 128 7 64 6 32 5 16 4 8 3 4 2 2 1 1 0 Cells Time

Time (sec) Cells

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 = red Exp = black Mathworld.wolfram.com

Hoffman-Singleton graph Mathworld.wolfram.com

Variation

Direct variation

Inverse variation