S 3.4 ECTION Linear Inequalities in Two Variables

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SECTION 3.4
Linear Inequalities in Two Variables
EXAMPLE 1
Graph
𝑥 + 4𝑦 ≥ −8.
EXAMPLE 2
Graph
𝑥 + 2𝑦 > 0.
EXAMPLE 3
Graph
𝑥 ≤ −3.
EXAMPLE 4
GRAPH 3𝑥 − 𝑦 ≥ 3 AND 𝑦 < 3.
EXAMPLE 5
GRAPH 6𝑥 − 4𝑦 < 10 OR 𝑦 > 2.
EXAMPLE 5
GRAPH 𝑦 ≤
1
𝑥 AND
3
𝑦 > −3𝑥 − 2.
SECTION 3.5
Introduction to Relations and Functions
DOMAIN & RANGE
EXAMPLE 1
Find the domain and range of the relation.
{(5,12.8), (10, 16.2), (15,18.9), (20, 20.7), (25, 21.8)}
FUNCTIONS
THE CONCEPT OF A FUNCTION
MORE ON FUNCTIONS


A fnc. is a relation in which no two ordered
pairs have the same first component and
different second components.
Four Representations
Verbally (in words)
 Numerically (table of values)
 Visually (a graph)
 Algebraically (a formula)

MAPPING – ILLUSTRATES HOW EACH MEMBER
OF THE DOMAIN IS PAIRED WITH EACH MEMBER
OF THE RANGE (NOTE: LIST DOMAIN AND
RANGE VALUES ONCE EACH, IN ORDER.)
Draw a mapping for the following.
(5, 1), (7, 2), (4, -9), (0, 2)
0
4
5
7
-9
1
2
Is this
relation a
function?
EXAMPLE 2
Determine whether each relation is a function and
give the domain and range:
A)
{(1,2), (3,4), (5,6), (5,8)}
B)
{(1,2), (3,4), (6,5), (8,5)}
ADDITIONAL EXAMPLE

Function?

D/R?
EXAMPLE 3
Determine if the given relation is a function and
give the domain and range.
a)
𝒙
𝒇(𝒙)
0
4
1
𝒙
𝒇(𝒙)
0
9
-1
1
-1
2
4
1
4
3
7
2
7
4
-2
3
-2
5
4
5
5
b)
EXAMPLE 3 (CONT.)
𝒙
𝒇(𝒙)
0
9
1
-1
1
4
2
7
3
-2
5
5
THE VERTICAL LINE TEST
y
y
5
5
x
x
-5
-5
-5
5
-5
5
VERTICAL LINE TEST



Graph the relation. (Use graphing
calculator or pencil and paper.)
Use the vertical line test to see if the
relation is a function.
Vertical line test – If any vertical line
passes through more than one point of the
graph, the relation is not a function.
GRAPHICALLY IDENTIFYING THE
DOMAIN & RANGE
Your domain is the 𝑥-values contained in the
graph.
 The 𝑦-values contained in the graph are
your range.
 Write domain and range using interval or
set-builder notation.

Interval Notation
Examples
−4, 5)
(−∞, 6
Set-Builder Notation
Examples
𝑥 −4 ≤ 𝑥 < 5
𝑦 𝑦≤6
Example 4
Determine if the graph is a function and state
the domain and
range.
y
y
5
5
a)
b)
x
-5
-5
5
x
-5
-5
5
Example 4 (cont.)
y
c)
d)
5
y
5
x
x
-5
-5
5
-5
-5
5
Example 4 (cont.)
y
e)
f)
5
y
5
x
-5
x
-5
-5
5
-5
5
A COMPARISON
y
y
5
5
x
-5
x
-5
-5
5
-5
D:
D:
R:
R:
5
SECTION 3.6
Function Notation and Linear Functions
EXAMPLE 1
Let 𝑓 𝑥 = −2𝑥 + 5 and 𝑔 𝑥 = −𝑥 2 − 𝑥 + 1. Calculate . . .
a.
𝑓 −2
1
−
2
b.
𝑔
c.
𝑔 2 − 𝑓 −1
d.
𝑔 𝜋
EXAMPLE 2
Let 𝑓 𝑥 = −2𝑥 + 5 and 𝑔 𝑥 = −𝑥 2 − 𝑥 + 1. Calculate . . .
a.
𝑔(𝑥 − 1)
b.
𝑓 𝑥+ℎ
EXAMPLE 3
Use the graph to find the following.
a.
𝑓 −2
b.
𝑓 0