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)

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• 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