Lesson 1 functions as models

GENERAL MATHEMATICS
CHAPTER 1: FUNCTIONS
Lesson 1: FUNCTIONS AS MODELS
Lesson 2: EVALUATING FUNCTIONS
Lesson 3: OPERATIONS ON FUNCTIONS
Prepared By:
ADDISON M. PASCUA
GRADE 11-HUMSS STUDENT

LESSON 1:
FUNCTIONS AS MODELS
LEARNING OUTCOME(S): At the end of the lesson, the learner is able to
represent life situations using functions, including piecewise functions.
LEARNING OUTLINES:
1. Relations and Functions
2. The Function as a Machine
3. Functions and Relations as a Table of
Values
4. Functions as Graph in the Cartesian Plane
5. Vertical Line Test
6. Functions as Representations of Real-Life
Situations
7. Piecewise Functions

RELATIONS AND FUNCTIONS
A relation is simply a set or
collection of ordered pairs.
Nothing really special about it.
An ordered pair, commonly
known as a point, has two
components which are the x and
y coordinates.

1. RELATION IN SET NOTATION.
2. RELATION IN TABLE. 3. RELATION IN GRAPH 4. RELATION IN MAPPING
DIAGRAM

The domain is the set of all x or input values. We may describe it
as the collection of the first values in the ordered pairs.
The range is the set of all y or output values. We may describe it
as the collection of the second values in the ordered pairs.

A function is actually a
“special” kind of relation
because it follows an extra
rule. Just like a relation, a
function is also a set of
ordered pairs; however, every
x-value must be associated
to only one y-value.

A RELATION THAT IS NOT A FUNCTION A RELATION THAT IS A FUNCTION
Since we have
repetitions or
duplicates of x-
values with different
y-values, then this
relation ceases to
be a function.
This relation is
definitely a function
because every x-
value is unique and
is associated with
only one value of y.

How about this
example
though? Is this
not a function
because we
have repeating
entries in x?

Yes, we have repeating
values of x but they are
being associated with the
same value of y. The point
(1,5) shows up twice, and
while the point (3,-8) is
written three times. This
table can be cleaned up by
writing a single copy of the
repeating ordered pairs.
The relation is now clearly a function!

Example 1: Is the relation expressed in the mapping diagram a function?
IT IS A FUNCTION
Each element of the domain is being traced
to one and only element in the range.
However, it is okay for two or more values
in the domain to share a common value in
the range. That is, even though the
elements 5 and 10 in the domain share the
same value of 2 in the range, this relation is
still a function.

IT IS A FUNCTION
There’s nothing wrong
when four elements
coming from the domain
are sharing a common
value in the range. This is a
great example of a function
as well.

The element 15 has two arrows
pointing to both 7 and 9. This is a
clear violation of the requirement to
be a function. A function is well
behaved, that is, each element in the
domain must point to one element in
the range. Therefore, this relation
is not a function.
IT IS NOT A FUNCTION

A single element in the domain is being
paired with four elements in the range.
Remember, if an element in the domain is
being associated with more than one
element in the range, the relation is
automatically disqualified to be a function.
Thus, this relation is absolutely not a
function.
IT IS NOT A FUNCTION

Example 5: Is the mapping diagram a relation, or function?
NEITHER A RELATION NOR A FUNCTION.
The element “2” in the domain is not
being paired with any element in the
range. Every element in the
domain must have some kind of
correspondence to the elements in the
range for it to be considered a relation,
at least. Since this is not a relation, it
follows that it can’t be a function.

RELATIONS FUNCTIONS
A relation is a rule that relates values from a
set of values called 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.
The elements of the domain can be imagined
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.
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-
values but different y-values.

THE FUNCTION AS A MACHINE
We can view a function as
something that can take an object
(as long as the object is in its
domain) and turn it into (or map it
to) a different object. We can
imagine it is some machine that
does this transformation. You put
some object into its input funnel. If
the input object fits into the funnel,
then the function machine will
process that object and turn it into
some other object, which comes
out its output chute.

FUNCTIONS AND RELATIONS AS A
TABLE OF VALUES
A table of values is a list of numbers that are
used to substitute one variable, such as within
an equation of a line and other functions, to
find the value of the other variable, or missing
number.

FUNCTIONS AS GRAPH IN THE
CARTESIAN PLANE
The vertical line test is a method that
is used to determine whether a given
relation is a function or not. The vertical
line test supports the definition of a
function. That is, every x-value of a
function must be paired to a single y-
value. If we think of a vertical line as an
infinite set of x-values, then intersecting
the graph of a relation at exactly one
point by a vertical line implies that a
single x-value is only paired to a unique
value of y.

CARTESIAN PLANE
If a vertical line intersects the graph in all places at exactly one
point, then the relation is a function.

CARTESIAN PLANE
If a vertical line intersects the graph in some places at more than one
point, then the relation is NOT a function.

FUNCTIONS AS REPRESENTATIONS OF
REAL-LIFE SITUATIONS
EXAMPLE:
Give a function C
that can represent
the cost of buying
x meals, if one
meal costs P40.
Solution:
Since each meal
costs P40, then
the cost function is
C(x)=40x.

FUNCTIONS AS REPRESENTATIONS OF
REAL-LIFE SITUATIONS
EXAMPLE:
A person is earning P600
per day to do a job.
Express the total Salary
(S) as a function of the
number (n) of days that the
person works.
ANSWER:
S(n)=600n

PIECEWISE FUNCTIONS
A user is charged P300 monthly for a particular mobile plan, which
includes 100 free text messages. Messages in excess of 100 are
charged P1 each. Represent the monthly cost for text messaging using
the function t(m), where m is the number of messages sent in a month.

PIECEWISE FUNCTIONS
A jeepney ride costs P8.00 for the first 4 kilometers and each additional
integer kilometer adds P1.50 to the fare. Use a piecewise function to
represent the jeepney fare in terms of the distance (d) in kilometers.

PIECEWISE FUNCTIONS
https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:absolute-value-piecewise-
functions/x2f8bb11595b61c86:piecewise-functions/v/evaluating-piecewise-functions-example
functions/x2f8bb11595b61c86:piecewise-functions/v/piecewise-function-example
functions/x2f8bb11595b61c86:piecewise-functions/e/evaluating-piecewise-functions
functions/x2f8bb11595b61c86:piecewise-functions/e/evaluate-step-functions-from-their-graph

ANSWER THE FOLLOWING
1. { 1,3 , 2,3 , 3,3 }
2.{ 2,3 , 2,3 , 4,3 }
3. { −3,3 , −2,3 , −5,3 }
4. { 1,4 , 2,4 , 3,4 }
YES
NO
YES
YES

SOURCES:
https://www.chilimath.com/lessons/intermediate-
algebra/relations-and-functions/
https://mathinsight.org/function_machine
https://sciencing.com/definition-table-values-
5142215.html
https://www.chilimath.com/lessons/intermediate-
algebra/vertical-line-test/
https://www.slideshare.net/lizamagalso/gen-math-g11-
introduction-to-functions?from_action=save

Lesson 1 functions as models

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