3. Definition of a Function:
β’ 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).
β’ Every value of x corresponds to a unique
value of y
GENERAL MATHEMATICS Samar College Galina V. Panela
5. Is it a function or not?
GENERAL MATHEMATICS Samar College Galina V. Panela
6. GENERAL MATHEMATICS Samar College Galina V. Panela
What is the difference
between a function and a
relation?
7. RELATIONS versus FUNCTIONS
GENERAL MATHEMATICS Samar College Galina V. Panela
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.
8. RELATIONS versus FUNCTIONS
GENERAL MATHEMATICS Samar College Galina V. Panela
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.
9. RELATIONS versus FUNCTIONS
GENERAL MATHEMATICS Samar College Galina V. Panela
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.
10. 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 Samar College Galina V. Panela
11. The function as a machineβ¦
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.
GENERAL MATHEMATICS Samar College Galina V. Panela
12. Determine if this machine produces
a functionβ¦
GENERAL MATHEMATICS Samar College Galina V. Panela
13. Determine if this machine produces
a functionβ¦
GENERAL MATHEMATICS Samar College Galina V. Panela
14. Determine if this machine produces
a functionβ¦
GENERAL MATHEMATICS Samar College Galina V. Panela
15. Determine if this machine produces
a functionβ¦
GENERAL MATHEMATICS Samar College Galina V. Panela
17. Table of Values
a
β’ A table of values is commonly
observed when describing a function.
β’ This shows the correspondence
between a set of values of x and a set
of values of y in a tabular form.
GENERAL MATHEMATICS Samar College Galina V. Panela
18. Examples of Table of Values
a
GENERAL MATHEMATICS Samar College Galina V. Panela
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
19. 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
GENERAL MATHEMATICS Samar College Galina V. Panela
21. Vertical Line Test
a
β’ 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 Samar College Galina V. Panela
22. Is this a function or not?
GENERAL MATHEMATICS Samar College Galina V. Panela
23. Is this a function or not?
GENERAL MATHEMATICS Samar College Galina V. Panela
24. Is this a function or not?
GENERAL MATHEMATICS Samar College Galina V. Panela
25. Relationship Between the Independent
and Dependent Variables
GENERAL MATHEMATICS Samar College Galina V. Panela
Input
(value of x)
Process
(equation
rule)
Output
(value of y)
26. Examples:
1. Find the value of y in the equation
y = 10x β 3 if x = - 5.
2. Find the value of x if the value of y
in the equation π² =
ππ+π
πβπ
is 2.
GENERAL MATHEMATICS Samar College Galina V. Panela
27. 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 π½ = π π
where s is the length of the
edge.
β’ What is the volume of the cube if the
length of the edge is 5 cm?
β’ What is the length of its edge if its
volume is 728 cubic meters?
GENERAL MATHEMATICS Samar College Galina V. Panela
29. Evaluating Functions
a
β’ It is the process of determining the
value of the function at the number
assigned to a given variable.
GENERAL MATHEMATICS Samar College Galina V. Panela
30. Example:
Let π(π) = π π
β ππ + π. Find the following
values of the function
a. f (2)
b. f (-1)
c. f (0)
d. f (- Β½ )
e. f (- 4)
GENERAL MATHEMATICS Samar College Galina V. Panela
31. Example:
Let π π± = ππ β π. 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 Samar College Galina V. Panela
32. Example:
Let h π± =
ππ+π
ππβπ
. Find the following values
of the function
a. h (1)
b. h (-2)
c. h (6)
d. h (0)
e. h (2)
GENERAL MATHEMATICS Samar College Galina V. Panela
33. REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS Samar College Galina V. Panela
DOMAIN AND RANGE OF
FUNCTIONS
34. Domain D of a Function
a
β’ It is the set of all x-coordinates in the
set of ordered pairs.
Range R of a Function
a
β’ It is the set of all y-coordinates in the
set of ordered pairs.
GENERAL MATHEMATICS Samar College Galina V. Panela
35. Determine the domain and the range
of the following:
a
GENERAL MATHEMATICS Samar College Galina V. Panela
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
36. More on Independent Variables
a
β’ There are instances in which not all
values of the independent variables
are permissible.
β’ That is, some functions have
restrictions.
GENERAL MATHEMATICS Samar College Galina V. Panela
37. Determine the domain and the range
of the following:
a π π =
π
π+π
π π = π π β ππ
π π =
π + π
π β π
GENERAL MATHEMATICS Samar College Galina V. Panela
38. Piece-wise Functions
a
β’ These are functions which are defined
in defined in different domains since
they are determined by several
equations.
GENERAL MATHEMATICS Samar College Galina V. Panela
39. Determine the domain and the range
of the following:
a π π =
π π =
GENERAL MATHEMATICS Samar College Galina V. Panela
{ 2x + 3 if x β 2
4 if x = 2
{ 2x + 3 if x < 1
β π π + π if x β₯ 1
41. Operations on Functions
If f and g are functions then
β’ (f + g) = f(x) + g(x)
β’ (f β g) = f(x)β g(x)
β’ (f β g) = f(x) β g(x)
β’
π
π
π =
π(π)
π(π)
where g(x) β 0
GENERAL MATHEMATICS Samar College Galina V. Panela
42. Example
Let f(x) = π π
β ππ + π and g(x)= x β 1.
Perform the operations and identify the
domain
β’ (f + g)
β’ (f β g)
β’ (f β g)
β’
π
π
π
GENERAL MATHEMATICS Samar College Galina V. Panela
43. Example
Let f(x)= x β 3 and g(x) = π π
+ π.
Perform the operations and identify the
domain
β’ (f + g)
β’ (f β g)
β’ (f β g)
β’
π
π
π
GENERAL MATHEMATICS Samar College Galina V. Panela
45. Operations on Functions
If f and g are functions then the
composite function denoted by π β π, is
defined by
π β π = π π(π)
GENERAL MATHEMATICS Samar College Galina V. Panela
46. Operations on Functions
The domain of π β π is the set of all
numbers x in the domain of g such that
g(x) is in the domain of f.
GENERAL MATHEMATICS Samar College Galina V. Panela
47. Example
Let f(x)= x β 3 and g(x) = π π
+ π. Find
β’ (π β π)(x)
β’ (π β π)(x)
β’ (π β π)(3)
β’ (π β π)(- 4)
GENERAL MATHEMATICS Samar College Galina V. Panela
49. Even and Odd Functions
β’ A function f is said to be even if
f(βx)=f(x) for each value of x in the
domain of f.
β’ A function f is said to be odd if
f(βx)= β f(x) for each value of x in the
domain of f.
GENERAL MATHEMATICS Samar College Galina V. Panela
50. Example
Determine whether each of the following
functions is even, odd or neither
β’ π π = ππ π
β ππ π
β ππ
β’ π π = βπ π
+ ππ π
β πππ
β’ π π = ππ π
β ππ π
β ππ β π
GENERAL MATHEMATICS Samar College Galina V. Panela