(GENERAL MATHEMATICS 11: LESSONS)

Lesson 1 - Introduction to Functions and its Operations

Lesson 2 - One-to-One & Inverse Functions
Lesson 3 - Rational Functions & Its Graph
Lesson 4 - Exponential & Logarithmic Functions
Additional Lesson 4 - Laws of Exponents & Logarithms
Lesson 5 - Simple & Compound Interest

➢ Functions and Operation on Functions

- A relation is a set of ordered pairs. The domain of a relation is the set of

first coordinates. The range is the set of second coordinates.

- A function is a relation in which each element of the domain corresponds to

exactly one element of the range. This is denoted by y = f (x).

Concept of Function
A function is a set of ordered pairs (x, y) such that no two ordered pairs have the same
x-value but different yvalues. Using functional notation, we can write f(x) = y, read as “f
of x is equal to y.” In particular, if (1, 2) is an ordered pair associated with the function f,
then we say that f(1)=2.

On the other hand, the vertical line test can be used to identify if the graph is a function.
A graph represents a function if and only if each vertical line intersects the graph at
most once.
Here are some real life examples of function:
1. Person’s salary depends on the number of hours worked
2. Buying rice in the market and its price per quantity
3. Internet Surfing fee and its hourly surfing rate

Function Rule
A function can be thought of as a rule, which operates on an input and produces an
output. This is often illustrated pictorially in two ways. The first way is by using a block
diagram, which consists of a box showing the input, output and the rule which, is often
written inside the box. The second way is to use two sets, one to represent the input
and one to represent the output with an arrow showing the relationship between them.
(ADDITIONAL EXAMPLES)
Piece-wise Function or Compound Function
- A functions having different values at different intervals and it composed of more
than one equation.

Real-life Applications/Situations:
● The price of airline tickets 5 months before the flight is constantly changing until
such time as the flight schedule comes nearer.
● Prices of utilities like water, electricity are differently charge per interval.
● Schools offer discounts depending on the number of sibling enrolling in the same
school.

(EXAMPLES)
A school’s fair committee wants to sell t-shirts for their school fair. They found a supplier
that sells t-shirts at a price of P175.00 a piece but can charge P15, 000 for a bulk order
of 100 shirts and P125.00 for each excess tshirt after that. Use a piecewise function to
represent the cost in terms of the number of t-shirts purchased.
The fee to park in the parking lot of a shopping mall costs P40.00 for the first two hours
and an extra P10.00 for each hour (or a fraction of it) after that. If you park for more than
twelve hours, you instead pay a flat rate of P200.00. Represent your parking fee using
the function p (t) where t is the number of hours you parked in the mall.

OPERATIONS ON FUNCTIONS
➢ One-to-One & Inverse Functions

One-to-One Function:
- A function is said to be one-to-one if and only if for each value of x, there
is at most one value of y and vice versa.

In graphing:
- A one-to-one function should satisfy both the vertical line test and the horizontal
line test.
Real-life situations that can be represented by one-to-one functions.

Example 1: Deoxyribonucleic acid is a molecule that carries most of the genetic

structures used in the development and functioning of all known living organisms and
microorganisms. It is the hereditary material in humans and almost all other organisms.
Every human being has a unique DNA molecule, and every existing DNA molecule is
unique to a human being. In other words, no two individual has exactly the same DNA
molecule.

Example 2: DepEd is developing a system of identification for all learners of the

Philippine public schools. This is the Learner’s Identification Number (LIS) System that
aims to provide a unique LIS to every public-school learner. Its aim is to ensure that no
two LIS is assigned two a Filipino learner, and that no to Filipino learners have the same
LIS number.

(Vertical Line Test)

- A test used to determine if a relation is a function. A relation is a FUNCTION if
there are NO vertical lines that intersect the graph at more than one point.

Identify whether each relation whose graphs are given is a function or not using
vertical line test.

(Function)

(Function)
 (Not a Function)

(Function)

(Not a Function)

(Horizontal Line Test)

- A function f is one-to-one if and only if no horizontal line drawn through the graph
of f intersect it more than once.

Identify whether each function whose graphs are given is a one-to-one or not
using horizontal line test.
 (One-to-One Function)

(Not One-to-One Function)

(One-to-one RELATION (since it’s not a

function)

(One-to-one function)
 (Not one-to-one relation)

Example: Show tables that represent and does not represent a one-to-one
functions.
Table 1 : 𝑓(𝑥) = 𝑥 + 1

For each y in the range, there is only one corresponding x. The graph passed horizontal
line test. Therefore, the function is a one-to-one function.

Example: 𝑓(𝑥) = |𝑥|

The graph passed the vertical line test; however, it did not pass the horizontal line test.
Therefore, the function is not a one-to-one function.

(Inverse Function)
−1
- The inverse of a one-to-one function 𝒇 is a one-to-one function 𝑓 (read as 𝒇
inverse), that consists of a set of all ordered pairs (𝑦, 𝑥), 𝑤ℎ𝑒𝑟𝑒 (𝑥, 𝑦) belong to 𝒇.

(Examples)
Finds the domain and range of each function and its inverse. Then, determine whether
the inverse is also a function or not.

a. 𝑔(𝑥) = {(1,17), (2,11), (3,5), (4, −1), (5, −7)}

b. 𝑓(𝑥) = {(0, −1), (2,5), (−5,1), (−4,5)}
SOLUTION:
a. 𝑔(𝑥) = {(17,1), (11,2), (5,3), (−1,4), (−7,5)} . The inverse of the function is still a
function.
b. 𝑓(𝑥) = {(−1,0), (5,2), (1, −5), (5, −4)}. The inverse of the function is not a function

2. Finding the inverse of a function.

Example: find the inverse of the function 𝑓(𝑥) = 2𝑥 + 1

Step 1: we know that 𝑓(𝑥) = 𝑦 𝑦 = 2𝑥 + 1

Step 2: interchange 𝑥 𝑎𝑛𝑑 𝑦 to find 𝑓 −1 𝑥 = 2𝑦 + 1
Step 3: solve for the new 𝑦 2𝑦 = 𝑥 − 1

𝑥−1
𝑦 = 2

Example: Verify whether the two are one-to-one and inverse function.

Function: (ordered pairs)

X -2 -1 0 1 2 3

Y 4 1 0 1 4 9
Inverse: (ordered pairs)

X 4 1 0 1 4 9

Y -2 -1 0 1 2 3

- Notice that although the original graph is a graph of a function, the resulting
graph is NOT A FUNCTION. That is why the one-to-one condition should be
satisfied before the inverse of a function can be obtained.

➢ Rational Functions & Its Graph

Rational Function:
- It is a function whose equation is a fraction, provided that the denominator and
𝑝(𝑥)
degree are not equal to 0. It is written in the form of 𝑓(𝑥) = 𝑞(𝑥)
- where: p(x) and q(x) are polynomial functions and q(x) ≠ 0. The domain of f(x) is
all values of x where q(x) ≠ 0.

Domain: It is the set of all real numbers except those that make the denominator q(x)
equal to zero or are all x values for which the function is defined.

Range: a collection of all real numbers that satisfy the rational function, f(x), or are all y
values for which the function is defined.

Intercepts: These are the points at which a graph crosses either the x or y-axis.

Y-intercept: The y-coordinate of a point where a line, curve, or surface intersects the
y-axis.

X-intercept: The x-coordinate of a point where a line, curve, or surface intersects the
x-axis Zeroes of Rational Functions: X value/s makes the function equal to zero.

Asymptotes: It is a line(s) that the graph keeps approaching but never touches.

Vertical Asymptote (VA): The vertical line(s) that the graph keeps approaching but
never touches- (x value/s)
Horizontal Asymptote (HA): The horizontal line(s) (y value/s)

Holes – A single point at which the function has no value or when a value of x sets both
the denominator and the numerator of a rational function equal to 0.

Steps in Graphing Rational Functions

Step 1: Find the intercepts, if there are any
Step 2: Find the Zeros of Rational Functions
Step 3: Find the vertical asymptotes (VA), horizontal asymptotes (HA), and the
intersection in HA, if there is any…
Step 4: Find the holes. Step 5: Sketch the graph

(RULES OF RATIONAL FUNCTIONS)

(EXAMPLE)
Find the domain, range, intercepts, asymptotes, the intersection in HA, and holes, if any,
and graph the following rational functions.

(DOMAIN) 𝑥−1=0𝑥=1

1. Determine which value of x in the All real numbers except 1 or

denominator makes the whole function (−∞, 𝟏) ∪ (𝟏, ∞) 𝒐𝒓 {𝒙 ∈ 𝑹|𝒙 ≠ 𝟏}
undefined by equating the denominator to
zero.
2. Write the answer in words or intervals
or set notation.
(RANGE) 𝑦 =
2𝑥
𝑥−1

1. Solve for the equation of x in the

function. 𝑥𝑦 − 𝑦 = 2𝑥
2. Identify which among the values of y in
the denominator makes the function 𝑥𝑦 − 2𝑥 = 𝑦
undefined by equating the denominator to 𝑥 (𝑦 − 2) 𝑦
zero. 𝑦−2
= 𝑦−2
3. Write the answer in words or intervals
or set notation. 𝑦
𝑥 = 𝑦−2
Note: (However, the range can only be
determined using more complex
mathematics, such as calculus, for some Equate the denominator: 𝒚 − 𝟐 = 𝟎 𝒚 = 2
rational functions.) Alternatively, you may
apply the Rules (THEOREM 7.2) to All real numbers except 2 or
identify the value of y that does not meet (−∞, 𝟐)𝑼(𝟐, ∞) 𝒐𝒓 {𝒚 ∈ 𝑹|𝒚 ≠ 𝟐}
the function.
Case 3: a= 2 , b = 1
CASE 1: degree of numerator < degree
of denominator y=0
2
𝑦 = 1
CASE 2: degree of numerator > degree
of denominator none 𝑦 = 2
CASE 3: degree of numerator = degree
of denominator y=a/b

(where a & b is the coefficient of the term

that has the highest degree)
 (STEPS) (PROCESS) (SOLUTIONS)

1. x-intercept x-intercepts:
• Simplify/Factor out
• Let y=0
• Solve for x
• Write the ordered
pair(answer)

2. y-intercept
y-intercept:
• Simplify/Factor out
• Let x =o
• Solve for y
• Write the ordered
pair(answer)

3. Zeroes • Simplify or factor out

• Let f(x)=y=0 or equate
the numerator to zero.
• Solve for x
Note: What would be your
x value in your xintercepts
that will be your zeroes.

4. Vertical Asymptote • Factor out but don’t

simplify (if necessary)
• Equate the denominator
into zero
• Solve for x (the value of x
will be the vertical
asymptote)
5. Horizontal Asymptote • Use the Cases in range.

6. Intersection in HA • Simplify or factor out, if

necessary.
• Substitute the value of
HA (y value).
• Write your answer in
ordered pair.

7. Holes • Equate both the Numerator:

numerator and the 2x=0
denominator to zero x=0
separately. If they have at
least one same root, then a Denominator:
hole exists. The common x-1=0
root is the xcoordinate.

• To find the y-coordinate, x=1

simplify the given function NO HOLES
and substitute the
x-coordinate.
Complete the table & graph

X -6 -4 -2 0 2 4 6

Y 1.7 1.6 1.3 0 4 2.7 2.4

➢ Exponential & Logarithmic Functions

Exponential Function
(Properties of exponential growth) (Properties of exponential decay)

● 𝑏>1 ● 0<𝑏<1
● The graph is increasing ● The domain is all real numbers
● The domain is all real numbers ● The graph is decreasing
● The graph increases without bound ● The graph increases without bound
as x approaches positive infinity as x approaches negative infinity
● The graph is continuous ● The graph is continuous
● The graph is smooth ● The graph is smooth

Exponential growth means that as the Exponential decay is when the uninfected
quantity increases, so does the rate at population gets lesser and lesser as the
which it grows) time passes by.

Determine the Domain, Range, Zeros, Intercepts, and Asymptotes of Exponential

Functions

Exponential Translation
Other types of exponential function graphs include transformations of functions derived
from an exponential function via a horizontal shift, vertical shift, or stretching.
(Horizontal Translation)

(Vertical Translation)

(Stretching Translation)

(More Examples)
(Let’s look at the graphs of both functions)

(When you put them on one cartesian plane, this is what it will look like)
(We can use several laws of exponents (called “rules of exponents”) to simplify
expressions that include numbers or variables raised to power.

(Examples of Converting Exponential and Logarithmic)

(Exponential Form) (Logarithmic Form)

4 𝑙𝑜𝑔2 (16) = 4
2 = 16

3 𝑙𝑜𝑔4 (64) = 3
4 = 64

3 𝑙𝑜𝑔7 (343) = 3
7 = 343

(Logarithmic Form) (Exponential Form)

𝑙𝑜𝑔6 (216) = 3 3
6 = 216

𝑙𝑜𝑔2 (0. 5) = 1 𝑜𝑟 1/2 −1

2 = 0. 5
 (DIFFERENT LAWS)

LAW 1: “The Product Rule” - 𝑙𝑜𝑔𝑏 𝑀 + 𝑙𝑜𝑔𝑏 𝑁

- (Logarithm of MN to the base of B equals logarithm of M to the base of B plus
logarithm of N to the base of B)
- The logarithm of a product is the sum of the logarithms)
Example: 𝑙𝑜𝑔4 (7𝑥)
Expand: 𝑙𝑜𝑔4 7 + 𝑙𝑜𝑔4 𝑥

LAW 2: “The Quotient Rule” - 𝑙𝑜𝑔𝑏 𝑚/𝑛 = 𝑙𝑜𝑔𝑏 𝑚 − 𝑙𝑜𝑔𝑏 𝑛

- Logarithm of m over n to the base of b equals logarithm of m to the base of b
minus logarithm of n to the base of b.
- The logarithm of a quotient is the difference of the logs.
Example: 𝑙𝑜𝑔𝑏 (𝑥/2)
Expand: 𝑙𝑜𝑔𝑏 𝑥 − 𝑙𝑜𝑔𝑏 2

𝑥
LAW 3: “The Power Rule” - 𝑙𝑜𝑔𝑏𝑀 = 𝑥 𝑙𝑜𝑔𝑏 𝑀
- Logarithm of M raised to the power of x to the base of b = x logarithm of M to the
base of b.
- The log of a number with an exponent is the product of the exponent and the log
of that number.
2
Example: 𝑙𝑜𝑔 𝑥
Expand: 2 𝑙𝑜𝑔 𝑥

Natural Logarithm - The natural logarithm of a number x is the logarithm to the base e,
where e is the mathematical constant approximately equal to 2.7182. It is usually written
using the short hand notation in x, instead of 𝑙𝑜𝑔𝑒 𝑥 as you expect.
Common Logarithms - Any logarithm with base 10.

Example: 𝑙𝑜𝑔10 𝑥 = 𝑙𝑜𝑔 𝑥

Change of base formula:

- Is used to write a logarithm of a number with a given base as the ratio of two
logarithms each with the same base that is different from the base of the original
logarithm.
- This helps us solve problems with logarithms that have different bases.

(Let a,b,x be positive real numbers, such that a ≠ 1 and b ≠ 1)

𝑙𝑜𝑔𝑎 𝑥
𝑙𝑜𝑔𝑏 𝑥 = 𝑙𝑜𝑔𝑎𝑏

➢ Simple & Compound Interest

(Definition of Terms)
● Interest: The amount of money you pay to borrow money or the amount of
money you earn on a deposit.
● Annual Interest Rate: The percent of interest that you pay for money borrowed
or earn for money deposit.
● Simple Interest: is calculated on the principal or original amount of the loan.
● Compound Interest: Interest that is earned on both the principal and any
previously earned interest.
● Maturity Value: The sum of the principal amount and the interest payments.

Annual Simple Interest:

The Simple Interest (𝐼𝑠 ), is calculated on the principal or original amount of the loan.

𝐼𝑠 = 𝑃𝑟𝑡
Where:
P – Principal
r – Rate
t – term or time in years

(Example)
Assume $ 250,000 is invested for five years at a simple interest rate of 10%.
Calculate the total amount and interest payable, at the end of the five-year period.

Given:
P = $ 250,000
r = 10% = 0.10
t = 5 years

Solution:
𝐼𝑠 = Prt = ($ 250,000)(0.10)(5 years)
𝐼𝑠 = $ 125,000

Maturity Value in a Simple Interest:

The future value 𝐹 is given by the formula: 𝐹 = 𝑃 (1 + 𝑟𝑡)

Where: P is the principal, r is the annual rate, and t is the term or time in years. Using
the problem in the Learning Content, the interest will be computed as shown.

P = $ 250,000 r = 10% = .10 t = 5 years

Solution:
F = P(1+rt)
= ($ 250,000)(1+(0.10)(5 years))
F = $ 375,000

(Example)
Assume your father deposited 10,000 Philippine peso in your BPI Family Savings bank
account at an annual rate of 1% compounded annually when you graduated from
kindergarten and you didn't receive the money until you finished grade 12. How much
money will you have in your bank account after 12 years?
Given: P = Php 10,000, r = 1% = 0.01 m = 1, t = 12 years

Solution:

𝑟 𝑚𝑡
𝐹 = 𝑃 (1 + 𝑚
)
0.01 1(2)
𝐹 = 10000 (1 + 1
)
𝐹 = 𝑃ℎ𝑝 11, 268. 25

(ADDITIONAL LECTURE VIDEOS)

Algebra Basics: What are functions?
Representations of Rational Functions
Graphing Rational Functions
One-to-One Functions
Inverse of One-to-One Functions

“Education is the most powerful weapon which you can use to change the world.”
-Nelson Mandela

-Steph S.
(G11 Nakaura - STEM Pre-Science)