Functions (Domain, Range, Composition). Symbols for Number Set Counting numbers ( maybe 0, 1, 2, 3,...

Functions (Domain, Range, Composition)

Symbols for Number Set

Counting numbers (maybe 0, 1, 2, 3, 4, and so on)

Natural Numbers:

Positive and negative counting numbers (-2, -1, 0, 1, 2, and so on)Integers:

a number that can be expressed as an integer fraction (-3/2, -1/3, 0, 1, 55/7, 22, and so on)

Rational Numbers:

a number that can NOT be expressed as an integer fraction (π, √2, and so on)

Irrational Numbers:NONE

Symbols for Number Set

The set of all rational and irrational numbers

Real Numbers:

Natural Numbers

Integers

Rational Numbers

Irrational

Numbers

Rea

l Num

ber

Ven

n D

iagr

am:

Set Notation

Not Included

The interval does NOT include the endpoint(s)

Interval Notation Inequality Notation GraphParentheses

( )

< Less than

> Greater than

Open Dot

Included

The interval does include the endpoint(s)

Interval Notation Inequality Notation GraphSquare Bracket

[ ]

≤ Less than

≥ Greater than

Closed Dot

Graphically and algebraically represent the following:

All real numbers greater than 11

Graph:

Inequality:

Interval:

Example 1

10 11 12

11x

11,Infinity never ends. Thus we always

use parentheses to indicate there is no

endpoint.

Describe, graphically, and algebraically represent

the following:

Description:

Graph:

Interval:

Example 2

1 3 5

1 5x

1,5

All real numbers greater than or equal to 1 and less than 5

Describe and algebraically represent the

following:

Describe:

Inequality:

Symbolic:

Example 3

-2 1 4

2 or 4x x

, 2 4,

All real numbers less than -2 or greater than 4

The union or combination of the

two sets.

Functions

Algebraic Function

Can be written as finite sums, differences, multiples, quotients, and radicals involving xn.

Examples:Transcendental

FunctionA function that is not Algebraic.

Examples:

A relation such that there is no more than one output for each input

A

B

C

W

Z

4

2

2 14

3 10xx

f x x x

g x

sin

ln

h x x

g x x

Domain and Range

DomainAll possible input values (usually x), which allows the function to work.

RangeAll possible output values (usually y), which result from using the function.

The domain and range help determine the window of a graph.

x y

f

Example 1

Domain: ,

Range: 25,

Domain: 8,2 2,9

Range: 7,8

1 9y x x

Describe the domain and range of both functions in interval notation:

Example 2

Sketch a graph of the function with the following characteristics:

1. Domain: (-8,-4) and Range: (-∞,∞)

2. Domain: [-2,3) and Range: (1,5)U[7,10]

Example 3

Find the domain and range of .

h t 4 3t

t -32 -20 -15 5 -4 0 1 2 3

h 10 8 7 -7 4 2 1 ER ER

0, DOMAIN: RANGE:

The range is clear from the graph and table.

The input to a square root function must be greater

than or equal to 0

4 3t 0

3t 4

t 43

Dividing by a negative switches

the sign

, 43

2 1

2 1

y yy

x x x

Slope Formula

The slope of the line through the points (x1, y1) and (x2, y2) is given by:

Forms of a Line

Point Slope Form - The equation of a line that contains the point (x1,y1) and whose slope is m is:

1 1y y m x x

Slope-Intercept Form - The equation of a line that contains the y-intercept (0,b) and whose slope is m is:

y mx b

General Form-

0Ax By C

Parallel and Perpendicular Lines

If the slope of line is then the slope

of a line…

• Parallel is

• Perpendicular is

am

b

am

b

bm

a

Example 1Algebraically find the slope-intercept equation of a

line that contains the points (-1,4) and (-4,-2).

2 1

2 1

y yx xm

2 44 1

63

2

4 2 1y x 4 2 2y x

1 1y y m x x 1 12y y x x 24 1y x

Find Slope

2 m

(-1,4)

(x1,y1)

(-4,-2)

(x2,y2)

2 6y x

Substitute into point-slope

Example 2Find an equation for the line that contains the point (2,-3)

and is parallel to the line .

2x y 6 0

Find the Slope of the original line:

2x y 6 0

Find the equation of the Parallel line:

Rewrite the equation into

Slope-Intercept

Form

m

Slope 2

y 2x 1

y 2x 6

y y1 m x x1 We know a

point and the slope

Parallel lines have same slope

y 2x 1

y 3 2 x 2

y 3 2x 4

Basic Types of Transformations

( h, k ): The Key Point

y a f x h k

When negative, the original graph is flipped about

the x-axis

When negative, the original graph is flipped

about the y-axis

Horizontal shift of h units

Vertical shift of k units

Parent/Original Function: y f xA vertical stretch if

|a|>1and a vertical

compression if |a|<1

Transformation Example1xy

Shift the parent graph four units to the left and three

units down.

Description:

14 3xy

Use the graph of below to describe and sketch the graph of .

Piecewise Functions

For Piecewise Functions, different formulas are used in different regions of the domain.

Ex: An absolute value function can be written as a piecewise function:

if 0

if 0

x xx

x x

Example 1Write a piecewise function for each given graph.

f x

f x

g x

g x

7

if x 4

5

if x 4

12 x 1 if 0x

x 1 if 0x

Example 2

Rewrite as a piecewise function.

f x x 2 1

Find the x value of the vertex

Change the absolute values to parentheses. Plus make the one on the bottom negative.

4-4

6

x -3 -2 -1 0 1 2 3 4

f(x) 6 5 4 3 2 1 2 3

f x

x 2 1

2 1x

if x 2

if x 2

Composition of Functions

g xff

g

First Second

f g xOR

Substituting a function or it’s value into another function. There are two notations:

(inside parentheses always first)

Example 1

Let and . Find:

1gf 2 5g x x 2 3f x x

211 5g

4

1 5

44 2 3f

11

8 3

1 11f g

Substitute x=1 into g(x) first

Substitute the result into f(x)

last

1gf

4

Example 2

Let and . Find:

g f x 2 5g x x 2 3f x x

2 3f x x

22 3 52 3g xx

24 12 9 5x x 2 3 2 3 5x x

24 12 4g f x x x

Substitute x into f(x) first

Substitute the result into g(x) last

24 12 4x x

24 12 9 5x x

g f x

2 3x