functions (domain, range, composition). symbols for number set counting numbers ( maybe 0, 1, 2, 3,...
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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