GENERAL MATHEMATICS 1.1 It is a special kind of relation wherein every value of x must
KEY CONCEPTS OF FUNCTIONS be associated with only one value of y.
● It is permissible to have multiple domain or x
Relation is a set of ordered pairs. The domain of a relation values that have the same y value or range, not the
is the set of first coordinates, x or input values. The range other way around.
is the set of second coordinates, y or output values. ● It is alright for 2 or more values in the domain to
Example of an ordered pair: share a common value in range.
x-coordinate → (8, -1) ← y-coordinate ● Each element in the domain must point to one
Examples of relations: element only in the range. (x = 11 & y = 7, 9)
1. {(1, 5), (2, 7), (3, 9), (4, 11)} ● If an element of the domain is associated with
2. {(1, a), (1, b), (1, c), (1, d)} more than one element in the range, the relation is
3. {(1, -1), (2, 0), (4, 2), (5, 3)} automatically disqualified to be a function.
4. {(1, √3), (2, √5), (8, √3)} ● Every element in the domain must have
Example: correspondence to the elements in the range.
In 2005, the population of cats in the infamous “Cat Island” Actually, this example (x = 1 & y = “nothing”) is also
of Japan is said to be 1, 500. In 2020, the population was not a relation since it is not an ordered pair.
determined to be 4, 300. Give the set of relations in the ● A table of values is a function if each element x
problem. corresponds to a unique element of y. However, it
Given: is merely a relation if there is at least one element
Year 2005 = 1, 500 population of x with more than corresponding y-value.
Year 2020 = 4, 300 population A relation that is NOT a function:
Set of Relation:
Year & Population = {(2005, 1500), (2020, 4300)} x y x y x y x y
Ways to represent relations: 1 5 -8 7 7 2 0 2
As long as the values are ordered pairs, then that becomes 2 7 -2 0 7 1 1
a relation. However, there are many ways to represent a
relation, not just in set notation. Set Notation is used to 2 9 0 4 7 3 2 1
essentially list numbers, objects or outcomes. Set notation
uses curly brackets { } which are sometimes referred to as 5 11 11 7, 9 7 -4 3 2
braces. Objects placed within the brackets are called the
elements of a set, and do not have to be in any specific A relation that is a function:
order.
x y x y x y
Example relation in set notation:
{(1, -1), (2, 0), (4, 2), (5, 3)} 1 -1 5 2 1 2
1. Table of values: It shows the correspondence
between a set of values of x and a set of values of 2 0 7 0 2 1
y in a tabular form (vertical or horizontal table).
3 1 8 1 3 3
2. Mapping: The elements of a domain are mapped
to the elements of the range using arrows. 4 2 10 2 4 -4
3. Graph: Each ordered pair is plotted as a point and
Vertical Line Test (VLT) is a graph which represents a
illustrated as a graph in the Cartesian Plane; a
function if and only if each vertical line intersects the
function passes the vertical line test.
graph at most once. A vertical line can be drawn anywhere
4. Domain & Range Descriptions / Ordered Pairs:
that intersects the graph in at most one point.
Shows a set of ordered pairs (x, y).
Function as a mathematical model can be written in the
Function is a relation in which each element of the domain
form y = f(x), where x is an independent variable.
corresponds to exactly one element of the range. The set
of all admissible values of x is called the domain and the A mathematical model is a function that represents
set of all the resulting values of y is called the range. relationships between two or more different quantities.
● x denotes independent variable
● y denotes dependent variable
, Types of Functions: Absolute Value function
Linear function The function f is an absolute value function, defined by
A function f is a linear function if f(x) = mx+b, where m and f(x)=│x│, if for all real numbers x, f(x)=x, for x≥0, -x, x≤0.
b are real numbers, and m and f(x) are not both equal to 0. The result of evaluating the absolute value function for any
nonzero value of x will always be positive. For example,
Example: y = 2x-3
f(-2)=│-2│=2 and f(2)=│2│=2.
● Using the slope-intercept form y = mx+b, we know
● y=│x│, opening upwards
that m=2 and b=-3.
● y=-│x│, opening downwards
● Linear equation graphs represent a line in the
Cartesian plane. Piecewise function
Linear Example: A piecewise function or a compound function is a function
defined by multiple sub-functions, where each sub-function
In 2005, the population of cats in the infamous “Cat Island”
applies to a certain interval of the main function’s domain.
of Japan is said to be 1, 500. In 2020, the population was
For example,
determined to be 4, 300. Calculate the cat population’s
average rate of change. f(x)=⎰8 if 0 < x ≤ 3
● Use the slope formula: 𝐦=𝐲₂-𝐲₁/𝐱₂-𝐱₁; rise over run ⎱2x+3/2 if x > 3
● The concept of slope is to measure the rate at Example 1: Fatima rides a jeepney to the Madrasah which
which changes are taking place. is 14km away from her house. The fare rate for the jeepney
ride is 9.00Php for the first 4km, and an additional of
Solution:
1.40Php for every km or a fraction of it thereafter.
m = 4300 - 1500 / (2020 - 2005) 1. What mathematical model can be drawn out from
= the given situation?
= 186.67 cats per year Kms x Fare f(x) Relationship
Quadratic function
First 4 9 f(4)=9
A quadratic function is any equation of the form
f(x)=ax²+bx+c, where a, b, and c are real numbers and a is 5(1+4) 9+1.4(5-4) f(5)=9+1.4(5-4)=10.4
not equal to 0.
Example: y = x² - 4x + 3 6(2+4) 9+1.4(6-4) f(6)=9+1.4(6-4)=11.8
● Find vertex first. Get the x-coordinate of the vertex x(x+4) 9+1.4(x-4) f(x)=9+1.4(x-4)=1.4x+3
by using the formula -b/2a. So, -(-4) / 2(1) which is .4
equal to 2. 2.
How much will she pay when she reaches her
● Substitute the value of x to the equation to get y. destination?
● y = (2)² - 4(2) + 3 x = 14kms
● y= 4-8 + 3 f(x) = 1.4x + 3.4
● y = -1 = 1.4(14) + 3.4
Constant function = 23Php
A linear function f is a constant function if f(x) = mx+b, Example 2: To sell more packs of dates, Lindsey needs to
where m=0 and b is any real number. This means that it will charge a lower price as indicated in the given table. The
always generate an output equal to b, no matter what input price for which Lindsey can sell x no. of packs is called the
value we give to it. price function f(x). This represents each data point in the
table.
Example:
1. f(x) = 2 Target no. of sales for Price per pack of dates
packs of dates (Php)
2. f(x) = 1, x ∈ [-1, 2]
Identity function 500 540
A linear function f is an identity function if f(x) = mx+b,
900 460
where m=1 and b=0. Thus, f(x) = x. Evaluating any value for
x will result in that same value. For example, f(0)=0 and 1, 300 380
f(2)=2.
1, 700 300
KEY CONCEPTS OF FUNCTIONS be associated with only one value of y.
● It is permissible to have multiple domain or x
Relation is a set of ordered pairs. The domain of a relation values that have the same y value or range, not the
is the set of first coordinates, x or input values. The range other way around.
is the set of second coordinates, y or output values. ● It is alright for 2 or more values in the domain to
Example of an ordered pair: share a common value in range.
x-coordinate → (8, -1) ← y-coordinate ● Each element in the domain must point to one
Examples of relations: element only in the range. (x = 11 & y = 7, 9)
1. {(1, 5), (2, 7), (3, 9), (4, 11)} ● If an element of the domain is associated with
2. {(1, a), (1, b), (1, c), (1, d)} more than one element in the range, the relation is
3. {(1, -1), (2, 0), (4, 2), (5, 3)} automatically disqualified to be a function.
4. {(1, √3), (2, √5), (8, √3)} ● Every element in the domain must have
Example: correspondence to the elements in the range.
In 2005, the population of cats in the infamous “Cat Island” Actually, this example (x = 1 & y = “nothing”) is also
of Japan is said to be 1, 500. In 2020, the population was not a relation since it is not an ordered pair.
determined to be 4, 300. Give the set of relations in the ● A table of values is a function if each element x
problem. corresponds to a unique element of y. However, it
Given: is merely a relation if there is at least one element
Year 2005 = 1, 500 population of x with more than corresponding y-value.
Year 2020 = 4, 300 population A relation that is NOT a function:
Set of Relation:
Year & Population = {(2005, 1500), (2020, 4300)} x y x y x y x y
Ways to represent relations: 1 5 -8 7 7 2 0 2
As long as the values are ordered pairs, then that becomes 2 7 -2 0 7 1 1
a relation. However, there are many ways to represent a
relation, not just in set notation. Set Notation is used to 2 9 0 4 7 3 2 1
essentially list numbers, objects or outcomes. Set notation
uses curly brackets { } which are sometimes referred to as 5 11 11 7, 9 7 -4 3 2
braces. Objects placed within the brackets are called the
elements of a set, and do not have to be in any specific A relation that is a function:
order.
x y x y x y
Example relation in set notation:
{(1, -1), (2, 0), (4, 2), (5, 3)} 1 -1 5 2 1 2
1. Table of values: It shows the correspondence
between a set of values of x and a set of values of 2 0 7 0 2 1
y in a tabular form (vertical or horizontal table).
3 1 8 1 3 3
2. Mapping: The elements of a domain are mapped
to the elements of the range using arrows. 4 2 10 2 4 -4
3. Graph: Each ordered pair is plotted as a point and
Vertical Line Test (VLT) is a graph which represents a
illustrated as a graph in the Cartesian Plane; a
function if and only if each vertical line intersects the
function passes the vertical line test.
graph at most once. A vertical line can be drawn anywhere
4. Domain & Range Descriptions / Ordered Pairs:
that intersects the graph in at most one point.
Shows a set of ordered pairs (x, y).
Function as a mathematical model can be written in the
Function is a relation in which each element of the domain
form y = f(x), where x is an independent variable.
corresponds to exactly one element of the range. The set
of all admissible values of x is called the domain and the A mathematical model is a function that represents
set of all the resulting values of y is called the range. relationships between two or more different quantities.
● x denotes independent variable
● y denotes dependent variable
, Types of Functions: Absolute Value function
Linear function The function f is an absolute value function, defined by
A function f is a linear function if f(x) = mx+b, where m and f(x)=│x│, if for all real numbers x, f(x)=x, for x≥0, -x, x≤0.
b are real numbers, and m and f(x) are not both equal to 0. The result of evaluating the absolute value function for any
nonzero value of x will always be positive. For example,
Example: y = 2x-3
f(-2)=│-2│=2 and f(2)=│2│=2.
● Using the slope-intercept form y = mx+b, we know
● y=│x│, opening upwards
that m=2 and b=-3.
● y=-│x│, opening downwards
● Linear equation graphs represent a line in the
Cartesian plane. Piecewise function
Linear Example: A piecewise function or a compound function is a function
defined by multiple sub-functions, where each sub-function
In 2005, the population of cats in the infamous “Cat Island”
applies to a certain interval of the main function’s domain.
of Japan is said to be 1, 500. In 2020, the population was
For example,
determined to be 4, 300. Calculate the cat population’s
average rate of change. f(x)=⎰8 if 0 < x ≤ 3
● Use the slope formula: 𝐦=𝐲₂-𝐲₁/𝐱₂-𝐱₁; rise over run ⎱2x+3/2 if x > 3
● The concept of slope is to measure the rate at Example 1: Fatima rides a jeepney to the Madrasah which
which changes are taking place. is 14km away from her house. The fare rate for the jeepney
ride is 9.00Php for the first 4km, and an additional of
Solution:
1.40Php for every km or a fraction of it thereafter.
m = 4300 - 1500 / (2020 - 2005) 1. What mathematical model can be drawn out from
= the given situation?
= 186.67 cats per year Kms x Fare f(x) Relationship
Quadratic function
First 4 9 f(4)=9
A quadratic function is any equation of the form
f(x)=ax²+bx+c, where a, b, and c are real numbers and a is 5(1+4) 9+1.4(5-4) f(5)=9+1.4(5-4)=10.4
not equal to 0.
Example: y = x² - 4x + 3 6(2+4) 9+1.4(6-4) f(6)=9+1.4(6-4)=11.8
● Find vertex first. Get the x-coordinate of the vertex x(x+4) 9+1.4(x-4) f(x)=9+1.4(x-4)=1.4x+3
by using the formula -b/2a. So, -(-4) / 2(1) which is .4
equal to 2. 2.
How much will she pay when she reaches her
● Substitute the value of x to the equation to get y. destination?
● y = (2)² - 4(2) + 3 x = 14kms
● y= 4-8 + 3 f(x) = 1.4x + 3.4
● y = -1 = 1.4(14) + 3.4
Constant function = 23Php
A linear function f is a constant function if f(x) = mx+b, Example 2: To sell more packs of dates, Lindsey needs to
where m=0 and b is any real number. This means that it will charge a lower price as indicated in the given table. The
always generate an output equal to b, no matter what input price for which Lindsey can sell x no. of packs is called the
value we give to it. price function f(x). This represents each data point in the
table.
Example:
1. f(x) = 2 Target no. of sales for Price per pack of dates
packs of dates (Php)
2. f(x) = 1, x ∈ [-1, 2]
Identity function 500 540
A linear function f is an identity function if f(x) = mx+b,
900 460
where m=1 and b=0. Thus, f(x) = x. Evaluating any value for
x will result in that same value. For example, f(0)=0 and 1, 300 380
f(2)=2.
1, 700 300
