Pre Calculus
Functions and Graphs
Functions• A function is a relation where each
element of the domain is paired with exactly one element of the range
• independent variable - x• dependent variable - y• domain - set of all values taken by
independent variable• range - set of all values taken by
the dependent variable
Mapping
3
-6
9
12
-1
5
0
-8
2
Representing Functions• notation - f(x)• numerical model - table/list of
ordered pairs, matching input (x) with output (y)
• US Prison Polulation (thousands)Year Total Male Female1980 329 316 131985 502 479 231990 774 730 441995 1125 1057 682000 1391 1298 932005 1526 1418 108
• graphical model - points on a graph; input (x) on horizontal axis … output (y) on vertical
• algebraic model - an equation in two variables
Vertical Line Test
Finding the range• implied domain - set of all real
numbers for which expression is defined
• example: Find the range 31
yx
31
yx
Continuity• http://www.calculus-help.com/tutor
ials
• function is continuous if you can trace it with your pencil and not lift the pencil off the paper
Discontinuities• point discontinuity
– graph has a “hole”– called removable – example
2 3 4
4x xf xx
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
5
4
3
2
1
0
-1
-2
A
• jump discontinuity - gap between functions is a piecewise function
• example 4, 2
1 , 2x xf x
x
• infinite discontinuity - there is a vertical asymptote somewhere on the graph
• example 2
22 3
12x xf xx x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
6
4
2
0
-2
-4
-6
Finding discontinuities• factor; find where function
undefined• sub. each value back into original
f(x)• results …
# infinite disc.0
0 point disc.0
Increasing - Decreasing Functions
• function increasing on interval if, for any two points
• decreasing on interval if
• constant on interval if
1 2 1 2 and , x x f x f x
1 2f x f x
1 2f x f x
Example:
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
8
6
4
2
0
-2
22f x x
Example:
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
5
4
3
2
1
0
-1
-2
-3
-4
2
2 1xg x
x
Boundedness of a Function
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
5
4
3
2
1
0
-1
-2
-3
-4
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
15
10
5
0
-5
-10
-15
Extremes of a Function• local maximum - of a function is
a value f(c) that is greater than all y-values on some interval containing point c.
• If f(c) is greater than all range values, then f(c) is called the absolute maximum
• local minimum - of a function is a value f(c) that is less than all y-values on some interval containing point c.
• If f(c) is less than all range values, then f(c) is called the absolute minimum
A
B
C
D
E
F
G
H
I
J
K
local maxima
Absolute maximum
Absoluteminimum local minima
Example: Identify whether the function has any local maxima
or minima
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
4 27 6f x x x x
Symmetry• graph looks same to left and right
of some dividing line• can be shown graphically,
numerically, and algebraically
• graph: 2f x x
x f(x)
-3 9 -1 1 0 0 1 1 3 9
numerically
algebraically• even function
– symmetric about the y-axix– example
f x f x
22 8f x x
• odd function– symmetric about the origin– example
f x f x
3 2f x x x
Additional examples: even / odd
2 4 5 3 2
3 6
2
3 1 2
f x x x y x x x
g x x f x x
Asymptotes• horizontal - any horizontal line
the graph gets closer and closer to but not touch
• vertical - any vertical line(s) the graph gets closer and closer to but not touch
• Find vertical asymptote by setting denominator equal to zero and solving
End Behavior• A function will ultimately behave
as follows:– polynomial … term with the highest
degree– rational function … f(x)/g(x) take
highest degree in num. and highest degree in denom. and reduce those terms
– example
4 3
5 25 7 8 1
6 2 5x x xf xx x