General Mathematics - Quarter 1 Module 1: Functions

General Mathematics - Quarter 1 - Module 1: Functions

Module Overview

This module thoroughly examines functions, focusing on their representation in various real-life situations and applications, with a detailed exploration of piecewise functions and how they model different scenarios.
It reinforces and expands upon students' prior knowledge of relations and functions acquired in Grade 8, providing a deeper understanding of these fundamental concepts.

What I Need to Know

This module is designed to enable students to achieve mastery over the core concepts of functions, emphasizing their practical application and relevance through real-life examples and problem-solving.
Learning Objectives:
1. Review and reinforce the foundational concepts of relations and functions, ensuring a solid understanding of their definitions and properties.
2. Define and thoroughly explain functional relationships, emphasizing their role as mathematical models that accurately represent and describe real-world situations.
3. Demonstrate the ability to represent a diverse range of real-life scenarios using functions, including the effective use of piecewise functions to model complex situations.

What I Know (Pre-Test)

Function Definition: A relation is considered a function when each element in the domain (input) is associated with only one unique value in the range (output).
- Answer: a. Function
Identifying Functions: Determine which of the given relations qualifies as a function, ensuring that each input has a single, distinct output.
- b. $g = {(3,2), (2,1), (8,2), (5,7)}$
Domain Definition: The domain of a relation is defined as the set of all possible x-values or inputs that can be used in the relation.
- Answer: c. Domain
Range Identification: Determine the range of a function from a given diagram, which includes identifying all possible output values.
- Answer: R:{a, b}
Function Representation in Tables: Identify which table accurately represents a function, ensuring each x-value corresponds to only one y-value.
- a.
- x 0 1 1 0
- y 4 5 6 7
Real-Life Functions: Identify real-life relationships that can be accurately represented as functions, ensuring each input has a unique output.
- c. The rule which assigns to each cellular phone unit to its phone number.
Non-Function Relations: Identify which relation does NOT meet the criteria of a function, indicating at least one input with multiple outputs.
- c. The rule which assigns religion to each person.
Function Representation (Salary): Express the total salary $S$ as a function of the number of days $n$ worked, given a fixed daily rate of ₱500.00.
- $S(n) = 500n$
- Answer: c. $S(n) = 500n$
Problem Solving (Typing Job): Calculate the total earnings for a typing job that includes a fixed daily rate plus an additional amount for each typing job completed.
- Fixed rate: ₱100
- Additional per job: ₱5.00
- For 5 typing jobs: $100 + (5 \times 5) = 125$
- Answer: c. ₱125.00
Fare Function:
- Answer: a. $f(x) = 100 + 5x$
Piecewise Function (Jeepney Fare): Represent the jeepney fare $F$ in terms of the distance $d$ traveled, using a piecewise function to account for different rates over different distances.
- ₱9.00 for the first 4 km, ₱0.75 for each additional km.
- Answer: d. F(d) = {9 \text{ if } 0 < d \le 4
Continuing Piecewise Function:
- Answer: d. F(d) = {(9 + 0.75(n) \text{ if } d > 4
Piecewise Function (Tax): Model a tax scenario using a piecewise function, with varying tax rates applied to different income brackets.
- First ₱30,000.00: 12% tax
- ₱30,000.00 - ₱50,000.00: 15% tax
- Over ₱50,000.00: 20% tax
- Answer: a. $t(x) = 0.12x \text{ if } x \le 30,000$
Continuing Piecewise Function:
- Answer: b. t(x) = 0.15x \text{ if } 30,000 < x \le 50,000
Continuing Piecewise Function:
- Answer: c. t(x) = 0.20x \text{ if } x > 50,000

Lesson 1: Representing Real-Life Situations Using Functions

What’s In

Relation: Defined as any set of ordered pairs, representing a connection between two sets of elements.
- Domain: The domain of a relation is the set containing all first elements of the ordered pairs, representing the input values.
- Range: The range of a relation is the set containing all second elements of the ordered pairs, representing the output values.
Function: A function is a specific type of relation where each element in the domain corresponds to exactly one element in the range, ensuring a unique output for each input.
Examples:
- $A = {(1,2), (2,3), (3,4), (4,5)}$ (Function)
- $B = {(3,3), (4,4), (5,5), (6,6)}$ (Function)
- $C = {(1,0), (0, 1), (-1,0), (0,-1)}$ (Not a function)
- $D = {(a,b), (b, c), (c,d), (a,d)}$ (Not a function)
Table of Values:
- Relations A and B are functions because each x-value corresponds to only one y-value, maintaining uniqueness.
- Table C is not a function, as the x-value -1 corresponds to two different y-values, 4 and 1, violating the function criteria.
Mapping Diagrams:
- Relations A and C are functions because each element in the domain is mapped to a unique element in the range, ensuring a single output for each input. Relation B is not a function because at least one element in the domain maps to more than one element in the range, indicating multiple outputs from a single input.
Vertical Line Test:
- The vertical line test is a graphical method to determine if a relation is a function. A graph represents a function if any vertical line drawn through the graph intersects it at only one point.
- Graphs A and C are functions because they pass the vertical line test, meaning no vertical line intersects the graph more than once. Graphs B and D are not functions because they fail the vertical line test, with at least one vertical line intersecting the graph at multiple points.
Functions can be represented in multiple ways, including through verbal descriptions (words), organized data in tables, visual mappings, algebraic equations, and graphical representations.

What’s New

Real-life scenarios illustrating the application of functions:

Soda Machine: When Mark inserts money and selects a button on a soda machine, the machine dispenses the corresponding soda. Here, the function rule is determined by the price of the soda. Input: money and button selection. Output: the selected soda.

Diameter-Circumference: Students measure the diameter and circumference of various circular objects. By dividing each circumference by its corresponding diameter, they observe a constant ratio, which approximates pi ( $\pi$ ). This illustrates a direct functional relationship between diameter and circumference.
Hourly Pay: Margareth earns ₱50.00 per hour working 5 days a week for 8 hours each day, totaling ₱2,000. This demonstrates a function where her salary depends on the number of hours worked. Additionally, it can illustrate a many-to-one relationship where multiple employees (domain) receive the same salary amount (range).
Tree Height: Students measure the shadow of a Narra tree using a meter stick. Functions are applied by using the ratio of the length of an upright ruler to its shadow to determine the height of the tree, demonstrating how ratios can help find unknown inputs in functional relationships.

What is It

*The scenarios presented effectively demonstrate functions through input and output relationships.

*In the soda machine example:

*the input (domain) consists of the money Mark uses combined with the button he selects

*the output (range) is the specific soda product dispensed

The diameter-circumference relationship illustrates a direct one-to-one correspondence, as each diameter corresponds to a unique circumference.

*Margareth's salary illustrates a many-to-one relationship.

*the employees (domain), including Margareth, receive the same salary amount (range), indicating multiple inputs mapping to a single output.

*Measuring the height of a tree involves applying a defined function rule.

*we use the same function rule (ratio) to compare the length of an upright ruler to its shadow, which helps determine the unknown height of the tree.

Function Machine:
- The function machine visually represents the input-output process of a function. An input is fed into the machine, processed according to the function's rule, and produces a specific output.
Equation Representation:
- The output (y) is dependent on the input (x), indicating that y is a function of x. This relationship is expressed through an equation where the value of y is determined by the value of x.
- Example: If a function machine always adds three (3) to whatever input is entered, the equation that represents this function is $x + 3 = y$ or $f(x) = x+ 3$ , where $f(x) = y$ . This equation shows how the input x is transformed to produce the output y.
Examples of Creating Equations:
- A. Express height (H) as a function of age (a), assuming an increase of 2 inches every year: $H(a) = 2 + a$
- B. Express distance (D) as a function of time (t), for a car traveling at a constant speed of 60 kilometers per hour: $D(t) = 60t$
- C. Express battery charge (B) as a function of hours (h), assuming a battery loses 12% of its charge per hour: $B(h) = 1 - 0.12h$
- D. Determine the volume of a box (V) created from a 10 in x 8 in rectangular sheet by cutting squares of side x from each corner: $V(x) = (10 - 2x)(8 - 2x)(x) = 80x - 36x^2 + 4x^3$
  *Piecewise Functions:
  *Piecewise functions are defined by multiple formulas, each applicable over a specific interval of the input domain. This allows for different rules or behaviors to be applied depending on the input value.
  * $f(x) = \begin{cases} \text{formula 1} & \text{if } x \text{ is in domain 1} \ \text{formula 2} & \text{if } x \text{ is in domain 2} \ \text{formula 3} & \text{if } x \text{ is in domain 3} \end{cases}$
Examples:
- A. Consider a mobile plan that charges ₱250.00 monthly for 200 free text messages, with each excess message charged at ₱1.00.
  - $t(m) = \begin{cases} 250 & \text{if } 0 < m \le 200 \ (250 + m) & \text{if } m > 200 \end{cases}$
- B. A chocolate bar is priced at ₱50.00 per piece, but the price reduces to ₱48.00 per piece for purchases of more than 5 pieces.
  - $f(n) = \begin{cases} 50 & \text{if } 0 < n \le 5 \ (48n) & \text{if } n > 5 \end{cases}$
- C. A catering service charges ₱250.00 per head for events with 50 or fewer attendees, ₱200.00 per head for 51-100 attendees, and ₱150.00 per head for events with over 100 attendees.
  $C(h) = \begin{cases} 250 & \text{if } n \le 50 \ 200 & \text{if } 51 \le n \le 100 \ 150 & \text{if } n > 100 \end{cases}$

What’s More

Independent Practice 1:
1. Express salary as a function of days worked, given a daily rate of ₱750.00: $S(n) = 750n$
2. Express jeepney fare as a function of kilometers traveled, with a base fare of ₱9.00 for the first 4 km and an additional ₱0.50 per additional km: $F(d) = 9 + 0.50(d - 4)$
Independent Assessment 1:
1. Express computer rental fee as a function of hours, charging ₱15.00 per hour: $R(t) = 15t$
2. Calculate the volume of a box formed by cutting squares of side $a$ from each corner of an 8 in x 6 in rectangle: $V(a) = (8 - 2a)(6 - 2a)(a) = 48a - 28a^2 + 4a^3$
  *Piecewise Functions:
3. Determine the tricycle fare, given a rate of ₱10.00 for the first 2 km and ₱8.00 per additional km.
  $C(d) = \begin{cases} 10 & \text{if } d \le 2 \ (10 + 8(d -2)) & \text{if } d \ge 3 \end{cases}$
4. Calculate the parking fee, with a charge of ₱25.00 for the first 2 hours, ₱5.00 per additional hour, and a flat rate of ₱100.00 after 12 hours.
  $p(t) = \begin{cases} 25 & \text{if } t \le 2 \ (25 + 5t) & \text{if } 3 \le t > 12 \ 100 & \text{if } t > 12 \end{cases}$
Independent Assessment 2:
1. A van rental costs ₱5,500.00 at a flat rate for up to 5 passengers, with an additional charge of ₱500.00 per additional passenger beyond 5.
- \begin{cases} 5500 & \text{if } n \le 5 \ (5500 + 500(n-5)) & \text{if } n > 5 \end{cases}
1. Calculate internet charges, with a rate of ₱500.00 for the first 30 GB, ₱30.00 per GB exceeding 30 GB, and a flat rate of ₱1,000.00 for 50 GB and above.
  - $C(g) = \begin{cases} 500 & \text{if } g \le 30 \ (500 + 30(g-30)) & \text{if } 31 \le g > 50 \ 1000 & \text{if } g \ge 50 \end{cases}$

What I Have Learned

A. Statement Analysis:
1. Incorrect. A relation is defined as a set of ordered pairs, where the first element is referred to as the domain, and the second element is known as the range.
  - Rewrite: A relation consists of ordered pairs, with the first element designated as the domain and the second as the range.
2. Incorrect (A function can be classified as one-to-one correspondence, one-to-many correspondence and many-to-one correspondence).
  Rewrite: Functions can be classified into types such as one-to-one correspondence and many-to-one correspondence, each describing different relationships between domain and range elements.
3. In a function machine, the input corresponds to the independent variable, while the output represents the dependent variable, illustrating how the output changes in response to the input.
B. Significance of Functions:
*Functions are essential for creating models of real-world phenomena and relationships, enabling predictions and problem-solving across various disciplines by expressing how one variable depends on another.
C. Piecewise Functions:
*Piecewise functions are utilized when varying conditions or rules apply across different intervals of the input domain, providing a more precise and adaptable representation of scenarios with diverse conditions.

What I Can Do

Identify three real-life situations that could be modeled as functions, formulate a sample problem for each, and derive the corresponding function equation to mathematically represent each situation.

Assessment (Post-Test)

Which statement is NOT true regarding functions?
*d. One-to-many correspondence is a function.*
In a relation, what term describes the y-values or the output?
*b. Range.*
Which of the following tables does NOT represent a function?
*d. x 0 2 4 6 y 6 5 4 3*
In this table, identify the domain of the function:
*x 1 2 3 4 5

y a b c d e

*C. D: {1, 2, 3, 4, 5}*

Which of the following relations are functions?
*c. h = {(6,1), (-2,3), (2, 6), (7, 2)}`*
Which of the following relations is a function?
*d. the rule which assigns each person a surname*
If a person can encode 1000 words per hour, which function expresses the total words W as a function of the number n of hours?
*c. $𝑊(𝑛) = 1000𝑛$ *
Judy earns ₱300.00 daily for cleaning Mrs. Perez’s house and an additional ₱25.00 per hour for taking care of Mrs. Perez’s child. Express Judy’s total salary (S) based on the time (h) spent caring for the child.
*a. $𝑆(ℎ) = 300 + 25ℎ$ *
Which function defines the volume of a cube?
*b. $𝑉 = 𝑠3$ , where s is the length of the edge*
With eighty meters of fencing, what function A represents the area that can be enclosed for Mang Gustin’s rectangular garden in terms of x?
*a. $𝐴(𝑥) = 40𝑥 − 𝑥2$ *
A mobile plan charges ₱400.00 monthly for 500 free texts, with each excess message charged at ₱1.00. Represent the monthly cost s(t) where t is the number of messages sent.
*a. 𝑠(𝑡) = {400, 𝑖𝑖𝑖𝑖 0 < 𝑡 ≤ 500*
A mobile plan charges ₱400.00 monthly for 500 free texts, with each excess message charged at ₱1.00. Represent the monthly cost s(t) where t is the number of messages sent.
*a. 𝑠(𝑡) = 400 + 𝑡, 𝑖𝑖𝑖𝑖 𝑡 > 500*
Cotta National High School GPTA offers t-shirts for the foundation day at ₱200.00 each, with a bulk order option of 100 shirts for ₱18,000.00 and ₱175.00 for each additional shirt. Use a piecewise function to express the cost $𝑡(𝑠)$ based on the number of shirts purchased.
*d. 𝑡(𝑠) = {200𝑠, 𝑖𝑖𝑖𝑖 0 < 𝑠 ≤ 99*
Cotta National High School GPTA offers t-shirts for the foundation day at ₱200.00 each, with a bulk order option of 100 shirts for ₱18,000.00 and ₱175.00 for each additional shirt. Use a piecewise function to express the cost $𝑡(𝑠)$ based on the number of shirts purchased.
*c. 𝑡(𝑠) = {18,000, 𝑖𝑖𝑖𝑖 𝑠 = 100*
Cotta National High School GPTA offers t-shirts for the foundation day at ₱200.00 each, with a bulk order option of 100 shirts for ₱18,000.00 and ₱175.00 for each additional shirt. Use a piecewise function to express the cost $𝑡(𝑠)$ based on the number of shirts purchased.
*a. 𝑡(𝑠) = {18,000 + 175(𝑠 − 100), 𝑖𝑖𝑖𝑖 𝑠 > 100*

Additional Activities

Model contaminated water with a pollutant concentration of 5 mg/L