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 Female

1980 329 316 13

1985 502 479 23

1990 774 730 44

1995 1125 1057 68

2000 1391 1298 93

2005 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/tutorials

• 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 x

f 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 , 2

x xf x

x

• infinite discontinuity - there is a vertical asymptote somewhere on the graph

• example 2

2

2 312

x xf x

x 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 1x

g xx

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

Absolute

minimumlocal 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 2

5 7 8 16 2 5x x x

f xx x