Download Lesson 4.3 Functions, Tables, Vertical Line Testing - Crest Ridge R-VII

Warm Up
Day 10 (8-21-09)
1. Evaluate (5k  3)  2k let k  3
2. Solve 3  (5 x  10)  2 x  4 x  (3x  7)
3. Graph x  7 or x  3
Informal Algebra II
Day 10 (8-21-09)
Objective:
1. Identify Domain and Range
2. Know and use the Cartesian Plane
3. Graph equations using a chart
4. Determine if a Relation is a Function
5. Use the Vertical Line Test for Functions
Relations
A
relation is a mapping, or pairing, of
input values with output values.
 The
set of input values is called the
domain.
 The
set of output values is called the
range.
Domain & Range
Domain is the set of
all x values.
Range is the set of all
y values.
Example 1: {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}
Domain- D: {1, 2}
Range- R: {1, 2, 3}
Example 2:
Find the Domain and Range of the
following relation:
{(a,1), (b,2), (c,3), (e,2)}
Domain: {a, b, c, e}
Range: {1, 2, 3}
Page 107
3.2 Graphs
Cartesian Coordinate System
 Cartesian
coordinate plane
 x-axis
 y-axis
 origin
 quadrants
Page 110
A Relation can be represented by a
set of ordered pairs of the form (x,y)
Quadrant II
X<0, y>0
Quadrant I
X>0, y>0
Origin (0,0)
Quadrant III
X<0, y<0
Quadrant IV
X>0, y<0
Plot:(-3,5) (-4,-2) (4,3) (3,-4)
(points
Every equation has solution points
which satisfy the equation).
3x + y = 5
Some solution points:
(0, 5), (1, 2), (2, -1), (3, -4)
Most equations have infinitely
many solution points.
Page 111
Ex 3. Determine whether the given ordered
pairs are solutions of this equation.
(-1, -4) and (7, 5); y = 3x -1
The collection of all solution points is
the graph of the equation.
Ex4 . Graph y = 3x – 1.
x
3x-1
y
Page 112
Ex 5. Graph
x
-3
-2
-1
0
1
2
3
y = x² - 5
x² - 5
y
What are your
questions?
3.3 Functions
•A relation as a function provided there is
exactly one output for each input.
•It is NOT a function if at least one input has
more than one output
Page 116
In order for a relationship to be a function…
EVERY INPUT MUST HAVE AN OUTPUT
TWO DIFFERENT INPUTS CAN HAVE THE
SAME OUTPUT
ONE INPUT CAN HAVE ONLY ONE
OUTPUT
INPUT
Functions
(DOMAIN)
FUNCTION
MACHINE
OUTPUT (RANGE)
Example 6
Which of the following relations are
functions?
R= {(9,10, (-5, -2), (2, -1), (3, -9)}
S= {(6, a), (8, f), (6, b), (-2, p)}
T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)}
No two ordered pairs can have the
same first coordinate
(and different second coordinates).
Identify the Domain and Range. Then
tell if the relation is a function.
Input
Output
-3
3
1
1
3
-2
4
Domain = {-3, 1,3,4}
Range = {3,1,-2}
Function?
Yes: each input is mapped
onto exactly one output
Identify the Domain and Range. Then
tell if the relation is a function.
Input
Output
-3
3
1
-2
4
1
4
Domain = {-3, 1,4}
Range = {3,-2,1,4}
Notice the set notation!!!
Function?
No: input 1 is mapped onto
Both -2 & 1
Look at example 1 on page 116
 Do
“Try This” a at the bottom of page 116
1. {(2,5) , (3,8) , (4,6) , (7, 20)}
2. {(1,4) , (1,5) , (2,3) , (9, 28)}
3. {(1,0) , (4,0) , (9,0) , (21, 0)}
The Vertical Line Test
If it is possible for a vertical line
to intersect a graph at more
than one point, then the graph
is NOT the graph of a function.
Page 117
Use the vertical line test to visually check if the
relation is a function.
(-3,3)
(4,4)
(1,1)
(1,-2)
Function?
No, Two points are on
The same vertical line.
Use the vertical line test to visually check if the
relation is a function.
(-3,3)
(1,1)
(3,1)
(4,-2)
Function?
Yes, no two points are
on the same vertical line
Examples
 I’m
going to show you a series of
graphs.
 Determine whether or not these
graphs are functions.
 You do not need to draw the graphs in
your notes.
#1
Function?
#2 Function?
#3 Function?
#4 Function?
#5
Function?
#6
Function?
#7 Function?
#8 Function?
#9 Function?
#10
Function?
#11
Function?
#12
Function?
Function Notation
f (x )
“f of x”
Input = x
Output = f(x) = y
Before…
Now…
y = 6 – 3x
f(x) = 6 – 3x
x
y
x
f(x)
-2
12
-2
12
-1
9
-1
9
0
6
0
6
1
3
1
3
2
0
2
0
(x, y)
(input, output)
(x, f(x))
Example 7
Find g(2) and g(5).
g = {(1, 4),(2,3),(3,2),(4,-8),(5,2)}
g(2) = 3
g(5) = 2
Example 8
Consider the function
h= { (-4, 0), (9,1), (-3, -2), (6,6), (0, -2)}
Find h(9), h(6), and h(0).
Example 9.
f(x) =
2
2x
–3
Find f(0), f(-3), f(5a).
Example 10.
F(x) = 3x2 +1
Find f(0), f(-1), f(2a).
f(0) = 1
f(-1) = 4
f(2a) = 12a2 + 1
Domain
The set of all real numbers that you
can plug into the function.
f :{( 3,0), ( 1,4), (0,2), (2,2), (4,1)}
D: {-3, -1, 0, 2, 4}
What is the domain?
Ex.
g(x) = -3x2 + 4x + 5
D: all real numbers
Ex.
x4
f ( x) 
x3
x+30
x  -3
D: All real numbers except -3
What is the domain?
Ex.
1
h( x ) 
x 5
x-50
D: All real numbers except 5
Ex.
f ( x) 
1
x2
x + 2 0
D: All Real Numbers except -2
What are your
questions?
Homework
 Page
108
14-20 even
 Page
114
4-32 (multiples of 4) (omit #8)
 Page
119
1-12 (yes or no)
14-28 even