Module 1
REVIEW ON FUNCTIONS
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What is a function?
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Definition of a Function:
• It is a relation define as a set of ordered
pairs (x, y) where no two or more distinct
ordered pairs have the same first element
(x).
• Every value of x corresponds to a unique
value of y
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Examples:
• Illustrations below are examples of a
function
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Is it a function or not?
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What is the difference
between a function and a
relation?
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RELATIONS versus FUNCTIONS
RELATIONS
FUNCTIONS
A relation is a rule that
relates values from a
set of values called the
domain to a second set
of values called the
range.
A function is a relation
where each element in
the domain is related to
only one value in the
range by some rule.
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RELATIONS versus FUNCTIONS
RELATIONS
FUNCTIONS
The elements of the
domain can be imagines
as input to a machine
that applies rule to
these inputs to generate
one or more outputs.
The elements of the
domain can be imagined
as input to a machine
that applies a rule so that
each input corresponds
to only one output.
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RELATIONS versus FUNCTIONS
RELATIONS
FUNCTIONS
A relation is also a set of A function is a set of
ordered pairs (x, y).
ordered pairs (x, y) such
that no two ordered pairs
have the same x-value
but different y-values.
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Is it a function or not?
a. f = {(0, -1), (2, -5), (4, -9), (6,-13)}
b. r ={(a, 0), (b, -1), (c, 0), (d, -1)}
c. g = (5, -10), (25, -50), (50, -100)
d. t = {(-2, 0), (-1, 1), (0, 1), (-2, 2)}
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The function as a machine…
We will try to represent mathematical
relations as machines with an input
and an output, and that the output is
related to the input by some rule.
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Determine if this machine produces
a function…
GENERAL MATHEMATICS
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Determine if this machine produces
a function…
GENERAL MATHEMATICS
Samar College
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Determine if this machine produces
a function…
GENERAL MATHEMATICS
Samar College
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Determine if this machine produces
a function…
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What is a table of
values?
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Table of Values
a
• A table of values is commonly
observed when describing a function.
• This shows the correspondence
between a set of values of x and a set
of values of y in a tabular form.
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Examples of Table of Values
a
x
y
0
-5
1
-4
4
-1
9
4
16
11
x
y
-1
-1
-1/4
- 1/2
0
0
1/4
1/2
1
1
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Is it a function or not?
1.A jeepney and its plate number
2.A student and his ID number
3.A teacher and his cellular phone
4.A pen and the color of its ink
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What is a vertical line
test?
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Vertical Line Test
a
• The vertical line test for a function
states that if each vertical line
intersects a graph in the x-y plane at
exactly one point, then the graph
illustrates a function.
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Is this a function or not?
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Is this a function or not?
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Is this a function or not?
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Relationship Between the Independent
and Dependent Variables
Input
(value of x)
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Process
(equation
rule)
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Output
(value of y)
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Examples:
1. Find the value of y in the equation
y = 10x – 3 if x = - 5.
2. Find the value of x if the value of y
𝟑𝒙+𝟖
in the equation 𝐲 =
is 2.
𝒙−𝟐
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Applications:
1.
2.
A car has travelled a distance of 124
kilometers in 4 hours. Find the speed of the
car.
The volume of the cube is defined by the
function 𝑽 = 𝒔𝟑 where s is the length of the
edge.
• What is the volume of the cube if the
length of the edge is 5 cm?
• What is the length of its edge if its
volume is 728 cubic meters?
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Module 1
REVIEW ON FUNCTIONS
EVALUATING FUNCTIONS
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Evaluating Functions
a
• It is the process of determining the
value of the function at the number
assigned to a given variable.
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Example:
Let 𝒇(𝒙) = 𝒙𝟐 − 𝟒𝒙 + 𝟒. Find the following
values of the function
a. f (2)
b. f (-1)
c. f (0)
d. f (- ½ )
e. f (- 4)
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Example:
Let 𝐠 𝐱 = 𝟑𝒙 − 𝟒 . Find the following
values of the function
a. g (2)
b. g (4)
c. g (0)
d. g (9)
e. g (- 1/3)
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Example:
𝟒𝒙+𝟖
.
𝟐𝒙−𝟒
Let h 𝐱 =
of the function
a. h (1)
b. h (-2)
c. h (6)
d. h (0)
e. h (2)
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Find the following values
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Module 1
REVIEW ON FUNCTIONS
DOMAIN AND RANGE OF
FUNCTIONS
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Domain D of a Function
a
• It is the set of all x-coordinates in the
set of ordered pairs.
Range R of a Function
a
• It is the set of all y-coordinates in the
set of ordered pairs.
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Determine the domain and the range
of the following:
a
x
y
0
-5
1
-4
4
-1
9
4
16
11
x
y
-1
-1
-1/4
- 1/2
0
0
1/4
1/2
1
1
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More on Independent Variables
a
• There are instances in which not all
values of the independent variables
are permissible.
• That is, some functions have
restrictions.
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Determine the domain and the range
of the following:
𝒇 𝒙 =
a
𝟓
𝒙+𝟐
𝒈 𝒙 = 𝒙𝟐 − 𝟔𝟒
𝒙+𝟑
𝒇 𝒙 =
𝒙−𝟐
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Piece-wise Functions
a
• These are functions which are defined
in defined in different domains since
they are determined by several
equations.
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Determine the domain and the range
of the following:
𝒇 𝒙 =
{
𝒇 𝒙 =
{
a
2x + 3 if x ≠ 2
4
if x = 2
2x + 3
– 𝒙𝟐 + 𝟐
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if x < 1
if x ≥ 1
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Module 1
REVIEW ON FUNCTIONS
OPERATIONS ON
FUNCTIONS
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Operations on Functions
If f and g are functions then
• (f + g) = f(x) + g(x)
• (f – g) = f(x)– g(x)
• (f ∙ g) = f(x) ∙ g(x)
•
𝒇
𝒈
𝒙 =
𝒇(𝒙)
𝒈(𝒙)
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where g(x) ≠ 0
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Example
𝟐
Let f(x) = 𝒙 − 𝟒𝒙 + 𝟑 and g(x)= x – 1.
Perform the operations and identify the
domain
• (f + g)
• (f – g)
• (f ∙ g)
•
𝒇
𝒈
𝒙
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Example
𝟐
Let f(x)= x – 3 and g(x) = 𝒙 + 𝟗 .
Perform the operations and identify the
domain
• (f + g)
• (f – g)
• (f ∙ g)
•
𝒇
𝒈
𝒙
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Module 1
REVIEW ON FUNCTIONS
COMPOSITE FUNCTIONS
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Operations on Functions
If f and g are functions then the
composite function denoted by 𝒇 ∘ 𝒈, is
defined by
𝒇 ∘ 𝒈 = 𝒇 𝒈(𝒙)
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Operations on Functions
The domain of 𝒇 ∘ 𝒈 is the set of all
numbers x in the domain of g such that
g(x) is in the domain of f.
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Example
Let f(x)= x – 3 and g(x) = 𝒙𝟐 + 𝟗. Find
• (𝒇 ∘ 𝒈)(x)
• (𝒈 ∘ 𝒇)(x)
• (𝒇 ∘ 𝒈)(3)
• (𝒈 ∘ 𝒇)(- 4)
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Module 1
REVIEW ON FUNCTIONS
EVEN AND ODD
FUNCTIONS
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Even and Odd Functions
• A function f is said to be even if
f(–x)=f(x) for each value of x in the
domain of f.
• A function f is said to be odd if
f(–x)= – f(x) for each value of x in the
domain of f.
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Example
Determine whether each of the following
functions is even, odd or neither
• 𝒇 𝒙 = 𝟒𝒙𝟒 − 𝟑𝒙𝟐 − 𝟏𝟎
• 𝒇 𝒙 = −𝒙𝟓 + 𝟑𝒙𝟑 − 𝟏𝟐𝒙
𝟑
𝟐
• 𝒇 𝒙 = 𝟒𝒙 − 𝟒𝒙 − 𝟖𝒙 − 𝟐
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