module1reviewonfunctions-160710051749.pptx

REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS

REVIEW
• 1. It is the first set of first coordinates
• A. Range
• B. Domain
• C. Abscissa
• 2. It is a set of second coordinates.
• A. Range
• B. Domain
• C. Abscissa

GENERAL MATHEMATICS
What is a function?

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
GENERAL MATHEMATICS

Examples:
• Illustrations below are examples of a
function
GENERAL MATHEMATICS

Is it a function or not?
GENERAL MATHEMATICS

GENERAL MATHEMATICS
What is the difference
between a function and a
relation?

RELATIONS versus FUNCTIONS
GENERAL MATHEMATICS
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.

GENERAL MATHEMATICS
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.

GENERAL MATHEMATICS
RELATIONS FUNCTIONS
A relation is also a set of
ordered pairs (x, y).
A function is a set of
ordered pairs (x, y) such
that no two ordered pairs
have the same x-value
but different y-values.

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)}
GENERAL MATHEMATICS

GENERAL MATHEMATICS
What is a table of
values?

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.
GENERAL MATHEMATICS

Examples of Table of Values
a
GENERAL MATHEMATICS
x 0 1 4 9 16
y - 5 - 4 - 1 4 11
x -1 -1/4 0 1/4 1
y -1 - 1/2 0 1/2 1

GENERAL MATHEMATICS
What is a vertical line
test?

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.
GENERAL MATHEMATICS

Is this a function or not?
GENERAL MATHEMATICS

Relationship Between the Independent
and Dependent Variables
GENERAL MATHEMATICS
Input
(value of x)
Process
(equation
rule)
Output
(value of y)

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
in the equation is 2.
GENERAL MATHEMATICS

Applications:
1. A car has travelled a distance of 124
kilometers in 4 hours. Find the speed of the
car.
2. The volume of the cube is defined by the
function 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?
GENERAL MATHEMATICS

REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS
EVALUATING FUNCTIONS

Evaluating Functions
a
• It is the process of determining the
value of the function at the number
assigned to a given variable.
GENERAL MATHEMATICS

Example:
Let . Find the following values of the
function
a. f (2)
b. f (-1)
c. f (0)
d. f (- ½ )
e. f (- 4)
GENERAL MATHEMATICS

Example:
Let . Find the following values of the
function
a. g (2)
b. g (4)
c. g (0)
d. g (9)
e. g (- 1/3)
GENERAL MATHEMATICS

Example:
Let h. Find the following values of the
function
a. h (1)
b. h (-2)
c. h (6)
d. h (0)
e. h (2)
GENERAL MATHEMATICS

REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS
DOMAIN AND RANGE OF
FUNCTIONS

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.
GENERAL MATHEMATICS

Determine the domain and the range
of the following:
a
GENERAL MATHEMATICS
x 0 1 4 9 16
y - 5 - 4 - 1 4 11
x -1 -1/4 0 1/4 1
y -1 - 1/2 0 1/2 1

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.
GENERAL MATHEMATICS

of the following:
a
GENERAL MATHEMATICS

Piece-wise Functions
a
• These are functions which are defined
in defined in different domains since
they are determined by several
equations.
GENERAL MATHEMATICS

of the following:
a
GENERAL MATHEMATICS
{ 2x + 3 if x ≠ 2
4 if x = 2
{ 2x + 3 if x < 1
– if x 1

REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS
OPERATIONS ON
FUNCTIONS

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)
• where g(x) ≠ 0
GENERAL MATHEMATICS

Example
Let f(x) = and g(x)= x – 1. Perform the
operations and identify the domain
• (f + g)
• (f – g)
• (f g)
GENERAL MATHEMATICS

Example
Let f(x)= x – 3 and g(x) = . Perform the
operations and identify the domain
• (f + g)
• (f – g)
• (f g)
GENERAL MATHEMATICS

REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS
COMPOSITE FUNCTIONS

If f and g are functions then the
composite function denoted by , is
defined by
GENERAL MATHEMATICS

The domain of is the set of all numbers
x in the domain of g such that g(x) is in
the domain of f.
GENERAL MATHEMATICS

Example
Let f(x)= x – 3 and g(x) = . Find
• )(x)
• )(x)
• )(3)
• )(- 4)
GENERAL MATHEMATICS

REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS
EVEN AND ODD
FUNCTIONS

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.
GENERAL MATHEMATICS

Example
Determine whether each of the following
functions is even, odd or neither
GENERAL MATHEMATICS

module1reviewonfunctions-160710051749.pptx

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module1reviewonfunctions-160710051749.pptx