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Real Numbers, Relations and
Functions
Pre-Calculus
Sangmi Amy Park 11A
Rational Numbers
• Counting Numbers are only positive integers
• Whole Numbers: Counting Numbers + 0
• Integers: Whole Numbers + negative integers
(which is …-3, -2, -1)
• Rational Numbers: terminating or repeating
decimals.
• Real Numbers that can’t be written as a ration of two
integers
Example 1
Prove that 0. 59 is rational number (if it is).
Set as 100𝑥
Set as 𝑥
This number is a
rational numbers.
0. 59 = 𝑥
59. 59 = 100𝑥
Because the fraction is
made of two integers.
100𝑥 − 𝑥 = 59. 59 − 0. 59
99𝑥 = 59
59
𝑥=
99
Irrational Numbers
• Cartesian
coordinate system
Example 1
• Fine the coordinate
of point A.
A lies 3 units left from
y-axis and 3 units up
from the x-axis.
The coordinate is (3,
3).
The domain is the set of all x values in the relation
domain = {1, 3, 5, 7}
This is a
based
on
relation
The next five slides are
{(3,
2), (1, 1), (5, 4), (7, 6)}
http://mathxtc.com/
range = {1, 2, 4, 6}
The range is the set of all y values in the relation
A relation assigns the x’s with
y’s
1
2
2
4
3
6
4
8
Domain (set of all x’s)
Range (set of all y’s)
This relation can be written {(1, 6), (2, 2), (3, 4), (4, 8)}
A function f from set A to set B is assigned to each
element x in the set A exactly one element y in the
set B.
1
2
3
4
5
This is a function.
Because each x
values are assigned
to each y value.
Set A is the domain
2
4
6
8
10
Set B is the range
This relation can be written {(1, 4), (2, 2), (3, 10), (4, 8), (5, 6)}
Each x can have only one y, but it CAN be the same y
as another x gets assigned to.
1
2
3
4
5
This is a function.
Because all the x
values are assigned
to one y value. And
one x value has one
y value.
Set A is the domain
2
4
6
8
10
Set B is the range
This relation can be written {(1, 6), (2, 6), (3, 6), (4, 6), (5, 6)}
Each x have only one y, but one x value assigned to two
y value.
1
2
3
Is it a
This is not a
function?
function.
2
4
6
8
10
One x value can be
assigned to only one
5
y value.
Set A is the domain
Set B is the range
4
Is the relation shown above a function?
No, because from
Set A 3 was
assigned both 2
and 10
This relation can be written {(1, 4), (2, 6), (3, 2), (3, 10), (4, 6), (5, 8)}
Is it a function or not? If it’s not, why not?
1
2
3
4
5
This is not a
function.
2
3 from Set A didn’t
get assigned to y
value.
Set A is the domain
4
6
8
10
Set B is the range
This relation can be written {(1, 4), (2, 6), (4, 8), (5, 2)}
Example 1
Given the relations
{(-1, -2), (-2, 3), (2, -3),
(3, 2)}, state its domain
and range.
Domain = {-1, -2, 2, 3}
Range = {-2, 3, -3, 2}
Example 2
Is this relation function?
1
2
3
4
5
This is a function.
Because all the x
value gets assigned
to one y value
Set A is the domain
2
4
6
8
10
Set B is the range
Example 3
(a): All x values are
(a)
assigned to y values,
so this graph is a
function
(b): Two x values are
(b)
assigned to one y
value, so this graph is
not a function
Which is/are Function(s)?
(c): All x values are
(c)
assigned to y values,
so this graph is a
function
(d):(d)Two x values are
assigned to one y
value, so this graph is
not a function
Name of function
𝒇 𝒙 =
Output number
Input number
𝟐
𝟐𝒙
− 𝟑𝒙 + 𝟔
Directions that tell you
what to do with input x in
order to produce the
corresponding output f(x)
Example 1
Find 𝑓(𝑘).
Replace x
into k.
𝒇(𝒙) = 𝟑𝒙𝟐 − 𝟓𝒙 + 𝟔
𝒇(𝒌) = 𝟑 𝒌 𝟐 − 𝟓 𝒌 + 𝟔
= 𝟑(𝒌𝟐 ) − 𝟓 𝒌 + 𝟔
𝟐
= 𝟑𝒌 − 𝟓𝒌 + 𝟔
Example 2
Find 𝑓(−3) .
Replace x
into -3.
𝒇(𝒙) = 𝟑𝒙𝟐 − 𝟓𝒙 + 𝟔
𝒇(−𝟑) = 𝟑 −𝟑
𝟐
− 𝟓 −𝟑 + 𝟔
= 𝟑 𝟗 − 𝟓 −𝟑 + 𝟔
= 𝟐𝟕 + 𝟏𝟓 + 𝟔
= 𝟒𝟖
Example 3
Find 𝑓(2).
Replace x
into 2.
𝒇(𝒙) = 𝟐𝒙𝟐 − 𝟑𝒙 + 𝟔
𝒇(𝟐) = 𝟐 𝟐 𝟐 − 𝟑 𝟐 + 𝟔
=𝟐 𝟒 −𝟑 𝟐 +𝟔
=𝟖−𝟔+𝟔
=𝟖
Example 4
Find f(4)+ f(-2).
f 𝒙 = 𝟑𝒙𝟐 − 𝟓𝒙 + 𝟔
f 4 = 3(4)2 −5 4 + 6
= 48 − 20 + 6 = 34
f −2 = 3(−2)2 −5 −2 + 6
= 12+10+6 = 28
𝑓 4 + 𝑓 −2 = 34 + 28 = 62