Activate Prior Knowledge 4: The Concept of Function

Activate Prior Knowledge 4
Relations abound in daily life: people are related to each other in many ways as parents and children,
teachers and students, employers and employees and many others. In addition, many quantities depend on
one or more changing variables. For instance; plant growth depends on sunlight and rainfall, speed
depends on distance traveled and time taken while voltage depends on current and resistance. These
situations, on the other hand, reflect the concept of functions in mathematics.
Relation and Function
A relation is a correspondence between two sets called the domain and the range such that to each
element of the domain, there is assigned one or more elements of the range.
Let A = {0,1,2} and B = {1,2,3} and let us say that an element x in A is related to an element y in B, if
and only if, x is less than y. Let us use the notation x R y as a shorthand for the sentences “x is related to
y”. Then
0R1 since 0<1
0R2 since 0<2
1R2 since 1<2.
On the other hand, if the notation x R y represents the sentence “x is not related to y”, then
1R1 since 1≮1
2R1 since 2≮1
THE CONCEPT OF FUNCTION

A function is a special type of relation. It is denoted by f(x) which is defined as a correspondence between
two sets such that to each element of the domain, there is assigned exactly one element of the range.
1. The Input – Output System
It may help to view a function as a machine. You put an input value into the machine and it gives you an
output value.
The system shows that if the input is 2, the output will be 5 given that the function is f(x) = 2x + 1. The
same goes for other values of x which will produce a unique output. The input values represent the
domain while the output values represent the range of the function.
Take note that all functions are relations but not all relations are functions. To determine if a relation is a
function, the following methods may be employed:
1.1. Mapping Diagram
Figure 1. The elements in the domain are D = {1, 2, 3, 4, 5} while the elements in the range are R = {d,
e, f}. Notice that two distinct elements of the domain are paired to exactly one element in the range, that
is 1 and 2 are paired to d and 4 and 5 are paired to f. In this case, the relation is called many-to-one. This
type of relation satisfies the definition of a function.
Figure 2. The elements in the domain are A = {1, 2, 3} while the elements in the range are B = {5, 6, 7}.
Obviously, each element in set A is paired to exactly one element in set B. This type of relation is
called one-to-one. This relation is considered as a function.
Figure 3. The elements in the domain are A = {1, 3, 5} while the elements in the range are B = {2, 3, 4,
5, 6}. In contrast to figure 1, one element in the domain is paired to two elements in the range.
Element 1 is paired to 2 and 3 while element 5 is paired to 5 and 6. This type of correspondence is called
one-to-many. This is a relation however this relation is NOT a function since an element in the domain is
already assigned to more than one element of the range.
1.2. Ordered Pair or Table of Values
A relation represented through ordered pairs or table of values (x,y) is recognized as a function if the x
values are not repeated.
The domain of Set A is represented by the x values D = {4, 5, 6, 7, 8} while the range consists of R =
{12, 15, 18, 21, 24}. Notice that the elements in the domain are distinct, which means that set A is a
function.
In contrast to set B, there are x values that are repeated which include 4 and 5. The domain of set B are D
= {4, 5, 6} while the range includes R = {12, 15, 18, 21, 24}. In this case, set B is NOT a function.
Set B is not a function but it is a relation which is one-to-many. We confirm this by applying the mapping
diagram.
In set A, you may have noticed 2 is repeated in the y column. Be mindful that the y values will not be our
concern, we always focus on the x values. Since there are no repeated values in x, we say that set A is a
function.
Set B also contains repeated values in y, however, this is not our concern. We always check the x values
and we notice that 1 is repeated. Thus, set B is NOT a function.
Graph
Since ordered pairs can be represented graphically, we identify if a graph is a function through
the vertical line test.
• If any vertical line passes through a graph at exactly one point, then the graph represents a
function.
Evaluating Functions
The modeling of infectious disease is an important mathematical tool that has been used to study the
mechanisms by which diseases spread, to predict the future course of an outbreak. Recently, a kind of
virus infection spread all over the world including the Philippines. The virus is popularly known as
Coronavirus 2019 (COVID – 19) which originated from Wuhan, China. Since December 31, 2019 and as
of July 12, 2020, the Word Health Organization has recorded 12, 698, 995 cases of COVID – 19 from
different affected countries.
Application of Equations and Functions
An equation expresses the equality of mathematical expressions. Each of the following is an equation:
8 + 5 = 13
4y – 6 = 10
x2 – 2x + 1 = 0
Each of the equations below is a first-degree equation in one variable. First degree means that the variable
has an exponent of 1.
x + 11 = 14 3z + 5 = 8z 2(6y – 1) = 34
A solution of an equation is a number that, when substituted for the variable, results in a true equation.
FIRST DEGREE-EQUATION
3 is a solution of the equation x + 4 = 7 because 3 + 4 = 7.

9 is NOT a solution of x + 4 = 7 because 9 + 4 ≠ 7.
In solving a first degree equation in one variable, the goal is to rewrite the equation with the variable
alone on one side of the equation and a constant term on the other side of the equation. The constant term
is the solution of the equation.
Example: Solve the equation: 5x + 9 = 23 – 2x
Solution:
1. There are no fractions or parenthesis. There are no like terms on either side of the equation. Use
Addition property to rewrite the equation with only one variable term. Add 2x to each side of the
equation.
5x + 9 = 23 – 2x
5x + 2x + 9 = 23 – 2x + 2x
7x + 9 = 23
7x + 9 – 9 = 23 – 9
7x = 14
x=2
The solution is 2.
Application of First Degree Equation

Example 1: Forensic Scientist have determined that the equation H = 2.9L + 78.1 can be used to
approximate the height H, in centimeters, of an adult on the basis of the length L, in centimeters, of
adult’s humerus (the bone extending from the shoulder to the elbow).
A. Use this equation to approximate the height of an adult whose humerus measures 36
centimeters.
B. According to this equation, what is the length of the humerus of an adult whose height is 168
centimeters.
Solution:
A. Substitute 36 for L in the given equation. Solve the resulting equation for H
H = 2.9L + 78.1
H= 2.9 (36) + 78.1
H = 104.4 + 78.1
H = 182.5
The adult’s height is approximately 182.5 centimeters.
B. Substitute 168 for H in the given equation. Solve the resulting equation for L.
H = 2.9L + 78.1
168 = 2.9L + 78.1
168 – 78.1 = 29.L + 78.1 – 78.1
89.9 = 2.9 L
31 centimeters = L
In many applied problems, we are not given an equation that can be used to solve problems. Instead, we
must use the given information to write an equation whose solution answers the question stated in the
problem.
Example: The cost of electricity in a certain city is Php 8 each of the first 300 kWh(Kilowatt-hours) and
Php 13 for each kilowatt-hour over 300 kWh. Find the number of kilowatt-hours used by a family that
receives a Php 519.5 electric bill.
Solution:
Let k = the number of kilowatt-hours used by the family. Write an equation and solve the equation for k.
Php 8.00 for each of the first 300 kWh + Php 13.00 for each kilowatt-hour over 300 = P 519.50
8(300) + 13 (k – 300) = 519.50
2400 + 13k – 3900 = 519.50
13k – 1500 = 519.50
13k – 1500 + 1500 = 519.50 + 1500
13 k = 2019.50
k = 155. 35
The family used 155.35 or approximately 156 kWh of electricity.

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Activate Prior Knowledge 4

Relation and Function

THE CONCEPT OF FUNCTION

1. The Input – Output System

Application of Equations and Functions

3 is a solution of the equation x + 4 = 7 because 3 + 4 = 7.

Application of First Degree Equation

The family used 155.35 or approximately 156 kWh of electricity.

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