Functions (Domain, Range, Composition)
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Transcript 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
fx
x
x1
0
3
Examples:
Transcendental
Function
gx
24xx41
A function that is not Algebraic.
hxsinx
Examples: gxlnx
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
t3
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 y2y1
x x2x1
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
y2y
1
x2x
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
2
x
2
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
yfx
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 y1
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 ifx0
Example 1
Write a piecewise function for each given graph.
gx
f x
5 if
x
fx
0
1 i
x
4
f x
gx 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
fgx
(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 fx
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