Download Functions (Domain, Range, Composition)

Functions (Domain, Range,
Composition)
Symbols for Number Set
Natural Numbers: Counting numbers
(maybe 0, 1, 2, 3, 4, and so on)
Integers: Positive and negative counting
numbers (-2, -1, 0, 1, 2, and so on)
Rational Numbers: a number that can be
expressed as an integer fraction
(-3/2, -1/3, 0, 1, 55/7, 22, and so on)
Irrational Numbers: a number that can
NOT be expressed as an integer
fraction (π, √2, and so on)
NONE
Symbols for Number Set
Real Numbers: The set of all rational
and irrational numbers
Real Number Venn
Diagram:
Rational Numbers
Integers
Irrational
Natural Numbers
Numbers
Set Notation
Not Included
The interval does NOT include the endpoint(s)
Interval Notation Inequality Notation Graph
Parentheses
( )
< Less than
> Greater than
Open Dot
Included
The interval does include the endpoint(s)
Interval Notation Inequality Notation Graph
Square Bracket
[ ]
≤ Less than
≥ Greater than
Closed Dot
Example 1
Graphically and algebraically represent the following:
All real numbers greater than 11
Graph:
10
Inequality:
Interval:
x
1
1
11,
11
12
Infinity never ends.
Thus we always
use parentheses to
indicate there is no
endpoint.
Example 2
Describe, graphically, and algebraically represent
the following:
1

x5
Description: All real numbers greater than or
equal to 1 and less than 5
Graph:
Interval:
1
1,5
3
5
Example 3
Describe and algebraically represent the
following:
-2
1
4
All real numbers less than -2 or
Describe:
greater than 4
Inequality:
Symbolic:
x


2
o
r
x

4
The union or
combination of the
two sets.


,2

4
,




Functions
A relation such that there is no more than one
output for each input
A
W
B
Z
C
Algebraic
Function
Can be written as finite sums, differences,
multiples, quotients, and radicals involving xn.
2
fx
x

x1
0
3
Examples:
Transcendental
Function
gx
 24xx41
A function that is not Algebraic.
hxsinx
Examples: gxlnx
Domain and Range
Domain
All possible input values (usually x),
which allows the function to work.
Range
All possible output values (usually
y), which result from using the
function.
f
x
y
The domain and range help determine the window of a graph.
Example 1
Describe the domain and range of both functions in
interval notation:
yx

1
x

9





,

Domain: 

8
,
2
2
,
9
Domain: 


2
5
,

Range: 

Range: 7,8
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

3
t



The input to a square root
function must be greater
than or equal to 0
4

3
t

0

3
t


4
4
Dividing by a
t3
negative switches
the sign

The range is
7
-7
4
2
1 ER ER
clear from


,  RANGE: 0, the graph
DOMAIN: 
t
-32 -20 -15
h
10
8
5
-4
0
1
4
3
2
3
and table.
Slope Formula
The slope of the line through the points (x1, y1)
and (x2, y2) is given by:

y y2y1


x x2x1
Forms of a Line
Slope-Intercept Form - The equation of a line that contains
the y-intercept (0,b) and whose slope is m is:
ym
xb

Point Slope Form - The equation of a line that contains the
point (x1,y1) and whose slope is m is:
y

y

m
x

x


1
1
General Form-
A
x

B
y

C

0
Parallel and Perpendicular Lines
a
If the slope of line is m 
then the slope
b
of a line…
• Parallel is
a
m 
b
• Perpendicular is
b
m 
a
Example 1
Algebraically find the slope-intercept equation of a
line that contains the points (-1,4) and (-4,-2).
(-4,-2)
Substitute into (x1,y1)
(x2,y2)
Find
Slope
y2y
1
x2x
1
m


24

4 1
 63
2
2 m
y

y

m
x

x


1
1
y

y

2
x
x


1
1
y

4

2
x


1


point-slope
y

4

2
x

1


y

4

22
x

y

2
x

6
Example 2
Find an equation for the line that contains the point (2,-3)
and is parallel to the line 2
.
x

y

6

0
Find the Slope of the original
line:

2
x

y

6

0
y


2
x

6
Find the equation of the Parallel
line:
We know a
point and the
slope
Rewrite the
equation into
m
Slope

Intercept
Slope


2
Form

y

y

m
x

x


1
1
y


3


2
x

2


Parallel
y

3


2
x

4
lines
y


2
x

1
have



y


2
x

1

same
slope
Basic Types of Transformations
Parent/Original Function:
When negative,
the original graph
is flipped about
the x-axis
yfx

A vertical stretch if
|a|>1and a vertical
compression if
|a|<1
Horizontal shift of h
units
y

a

f

x

h

k


When negative, the
original graph is flipped
about the y-axis
Vertical shift of k units
( h, k ): The Key Point
Transformation Example
Use the graph of y1
x below to describe and
1
sketch the graph of y
.


3
x

4
Description:
Shift the parent
graph four units to
the left and three
units down.
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:

xifx0

x
x ifx0

Example 1
Write a piecewise function for each given graph.
gx
f x

5 if
x
fx

0
1 i
x


4
f x
gx  1
fx

0


1i
7 if
x


4
2x



Example 2
Rewrite f
as a piecewise function.
x

x

2

1


6
x
-3
f(x) 6

-2
-1
0
1
2
3
4
5
4
3
2
1
2
3
Find the x value of
the vertex
-4
Change the absolute
values to parentheses.
Plus make the one on
the bottom negative.
4


2
1if

x

x

2
f x
x

2

1if

x

2

Composition of Functions
Substituting a function or it’s value into another
function. There are two notations:
f gx

Second
OR
First
fgx
(inside parentheses always first)
g
f
Example 1
2
Let f
and
. Find:
g
x


x
5
x
2
x3
 
f g
1
42

4

3
g
1

1
5 f







2
Substitute
x=1 into
g(x) first


1
5


83
4
4
11
f
g
1
1
1



Substitute
the result
into f(x)
last
Example 2
2
Let f
and
. Find:
g
x


x
5
x
2
x3
 
g fx
x

Substitute the result into g(x) last
g
2
x

3


2
x

3

5





2
x

3
2
x

3

5




f
x
2
x
3
2
x
3
2


4
x
2
x

9

5
 1

2


4
xx

1
2

9

5
2


4
x

1
2
x

4
2
g
f
x


4
xx
1
2

4



 
Substitute x
into f(x) first
2