Download Relations and Functions

Lesson 1.1
Relations and
Functions
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A relation
is a pairing of elements of one
set with elements of a second set.
A domain
is a set of the first elements of
ordered pairs or a set of
abscissas.
A range
A function
is a set of the second elements of
ordered pairs or a set of ordinates.
is a relation in which each element of
the domain is paired with exactly one
element in the range.
Example 1.
State the domain and range of each relation. Then state whether
the relation is a function.
a. {(-1, 2), (0, 4), (1, 3)}
The domain is {-1, 0, 1}
The range is {2, 3, 4}
domain
range
-1
2
0
4
1
3
This relation is a function, since each element of the
domain is paired with exactly one element of the range.
b. {(-3, 1), (-2, 5), (-3, 4), (1, 2)}
The domain is {-3, -2, 1}
The range is {1, 2, 4, 5}
domain
range
1
-3
2
-2
4
1
5
This relation is not a function, since -3 is paired with two
elements of the range.
Activity 3.
The domain of a relation is consecutive even integers between and
including -4 and 4. The range, y, of the relation is three more than
half x, where x is a member of the domain. Write the relation as
a table of values and as an equation. Then graph the relation.
Graph:
Table:
Equation:
x
y
-4
1
-2
2
0
3
2
4
4
5
Activity 4.
OCEANOGRAPHY The table below shows the pounds per square
inch (psi) of pressure that push on a body at various water depths.
State the relation of the data as a set of ordered pairs. Also, state
the domain and range of the relation.
Water Pressure
Depth (ft)
Pressure (psi)
0
14.7
16.5
22.05
33
29.4
49.5
36.75
66
44.1
82.5
51.45
99
58.8
115.5
66.15
Relation: {(0, 14.7), (16.5, 22.05),
(33, 29.4), (49.5, 36.75), (66, 44.1),
(82.5, 51.45), (99, 58.8), (115.5, 66.15)}
Domain: {(0, 16.5, 33, 49.5, 66, 82.5,
99, 115.5}
Range: {14.7, 22.05, 29.4, 36.75, 44.1,
51.45, 58.8, 66.15}
Activity 6
State the domain and range of each relation.
a)
Domain: all real numbers.
Range: all real numbers.
b)
Domain: all real numbers.
Range: all negative numbers and
zero.
Activity 8
Vertical Line Test.
If every vertical line drawn on the graph of a relation
passes through no more than one point of the graph, then
the relation is a function.
Determine if the graph of each relation represents a
function. Explain.
a)
No, the graph fails the vertical line test.
b)
The graph represents a function since it
passes the vertical line test.
Activity 10.
Function notation is used to denote a function. In
function notation, the symbol f(x) is read "f of x" and
should be interpreted as the value of the function
f at x.
y=f(x)
y=2x then f(x)=2x
Evaluate each function for the given value.
a. f(-2) if f(x) = 2x4 - 3x3 + 4x - 1
b. g(3) if g(x) = |5 - x3|
f(-2) = 2(-2)4 - 3(-2)3 + 4(-2) - 1
g(3)= |5 - (3)3|
= 32 + 24 - 8 - 1
= 47
=|5 - 27|
= |-22|
=22
Activity 12.
a.
h(b)
if h(x) = 3x4 - 2x3 + x2 - x
h(b)= 3(b)4 - 2(b)3 + (b)2 - b
x=b
= 3b4 - 2b3 + b2 - b
b.
j(2a - 1)
if j(x) = 2x2 - 5x + 1
j(2a - 1)= 2(2a - 1)2 - 5(2a - 1) + 1
= 2(4a2 - 4a + 1) - 5(2a - 1) + 1
= 8a2 - 8a + 2 - 10a + 5 + 1
= 8a2 - 18a + 8
x = 2a - 1
Activity 14.
a.
f(x) =
x2
2
x  6x  8
To determine the included values
solve.
x2 - 6x + 8 = 0
(x - 2)(x - 4) = 0
x = 2 or x = 4
Therefore, the domain includes all real numbers
except 2 and 4.
b.
g(x) =
9
x3
Any value that makes the radical
negative must be excluded
from the domain of g since the square root
of a negative number is not a real number.
Also, the denominator cannot be zero.
Let x + 3 > 0 and solve it to determine included values.
x + 3> 0
x>-3
The domain is {x | x > -3}, which is
read “the set of all x such that
x is greater than -3.”