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
