General Mathematics - Functions

General Mathematics - Quarter 1 - Module 1: Functions

Module Overview

  • This module focuses on functions, particularly representing them in real-life situations, including piecewise functions.
  • It builds upon prior knowledge of relations and functions from Grade 8.

What I Need to Know

  • The module aims to help master key concepts of functions, especially representing them in real-life contexts.
  • Specific objectives:
    • Recall the concepts of relations and functions.
    • Define and explain functional relationships as mathematical models of situations.
    • Represent real-life situations using functions, including piecewise functions.

What I Know (Pre-Test)

  1. Function Definition: A relation where each element in the domain is related to only one value in the range.
    • Answer: a. Function
  2. Identifying Functions: Determine which relation is a function.
    • Correct option: b. g = {(3,2), (2,1), (8,2), (5,7)}
  3. Domain: Set of x values or inputs in a relation.
    • Answer: c. Domain
  4. Range: Determining the range of a function from a diagram.
    • Answer: b. R:{a, b}
  5. Function Representation in Tables: Identify which table represents a function.
    • Answer: d.
  6. Real-Life Functions: Identify real-life relationships that represent a function.
    • Answer: c. The rule which assigns to each cellular phone unit to its phone number.
  7. Non-Functions: Identifying relations that are NOT functions.
    • Answer: c. The rule which assigns religion to each person.
  8. Function Representation: Expressing total salary SS as a function of the number of days nn worked, earning ₱500.00 per day.
    • S(n)=500nS(n) = 500n
    • Answer: c.
  9. Problem Solving with Functions: Johnny earns ₱100 a day plus ₱5.00 per typing job. How much for 5 typing jobs?
    • 100+(55)=125100 + (5 _ 5) = 125
    • Answer: c. ₱125.00
  10. Fare Function: Find the fare function f(x)f(x) where xx is the number of typing jobs, with a fixed rate of ₱100 and ₱5 per job.
    • f(x)=100+5xf(x) = 100 + 5x
    • Answer: a.
  11. Piecewise Function - Jeepney Fare (1): Lucena jeepney fare is ₱9.00 for the first 4 km.
    • F(d)=9F(d) = 9 if 0 < d ≤ 4
    • Answer: d.
  12. Piecewise Function - Jeepney Fare (2): Additional ₱0.75 per km after the first 4 km. Use nn for additional kilometers.
    • F(d)=9+0.75(n)F(d) = 9 + 0.75(n) if d > 4
    • Answer: d.
  13. Piecewise Function - Tax Law (1): First ₱30,000.00 taxed at 12%.
    • t(x)=0.12xt(x) = 0.12x if x≤30,000x ≤ 30,000
    • Answer: a.
  14. Piecewise Function - Tax Law (2): Earnings > ₱30,000.00 and up to ₱50,000.00 taxed at 15%.
    • t(x)=0.15xt(x) = 0.15x if 30,000 < x ≤ 50,000
    • Answer: b.
  15. Piecewise Function - Tax Law (3): Earnings > ₱50,000.00 taxed at 20%.
    • t(x)=0.20xt(x) = 0.20x if x > 50,000
    • Answer: c.

Lesson 1: Representing Real-Life Situations Using Functions

What’s In
  • Relation: Any set of ordered pairs.
    • Domain: Set of all first elements of the ordered pairs.
    • Range: Set of all second elements of the ordered pairs.
  • Function: A relation where each element in the domain corresponds to exactly one element in the range.
  • Examples:
    • A=(1,2),(2,3),(3,4),(4,5)A = {(1,2), (2,3), (3,4), (4,5)} (Function)
    • B=(3,3),(4,4),(5,5),(6,6)B = {(3,3), (4,4), (5,5), (6,6)} (Function)
    • C=(1,0),(0,1),(−1,0),(0,−1)C = {(1,0), (0,1), (-1,0), (0,-1)} (Not a function - 0 has two outputs)
    • D=(a,b),(b,c),(c,d),(a,d)D = {(a,b), (b,c), (c,d), (a,d)} (Not a function - a has two outputs)
  • Tables: Identifying functions from tables of values.
    • Tables A and B represent functions because each xx value corresponds to exactly one yy value.
    • Table C is not a function because -1 corresponds to two values, 4 and 1.
  • Diagrams/Mappings: Identifying functions from mapping diagrams.
    • Diagrams A and C are functions because each element in the domain corresponds to a unique element in the range.
    • Diagram B is a relation but not a function because one domain element corresponds to multiple range elements.
  • Graphs: Using the vertical line test to identify functions.
    • If any vertical line drawn through the graph intersects it at exactly one point, the graph represents a function.
    • Graphs A and C are functions; graphs B and D are not.
  • Representations of Functions: Functions can be represented through words, tables, mappings, equations, and graphs.
What’s New
  • Real-life scenarios depicting functions:
    1. Soda Machine: Input (money + button), Output (soda). Function rule: product price. This shows one-to-one correspondence.
    2. Diameter and Circumference: Finding the relationship between diameter and circumference. Function rule: C=πdC = \pi d.
    3. Salary as a Function: Margareth earns ₱50.00 per hour, working 8 hours a day for 5 days, receiving ₱2,000 per week. This shows a many-to-one relationship between employees and salary.
    4. Height of a Tree: Measuring the shadow of a tree to find its height. Functions are applied using ratios.
What is It
  • Functions as representations of real-life situations
    • Function Machine: Illustration of functions with inputs and outputs. If an orange is put into a juicer, an orange juice output is expected, and not a grape juice.
    • Output (y) is dependent on input (x), so yy is a function of xx.
    • Example: If a function machine adds 3 to any input, the equation is f(x)=x+3f(x) = x + 3.
  • Creating Equations from Situations:
    • A. Height (H) as a function of age (a), with height increasing by 2 inches each year: H(a)=2+aH(a) = 2 + a.
    • B. Distance (D) as a function of time (t), with a car traveling 60 km per hour: D(t)=60tD(t) = 60t.
    • C. Battery charge (B) as a function of hours (h), with a 12% battery loss per hour: B(h)=1−0.12hB(h) = 1 - 0.12h.
    • D. Volume of a box formed by cutting squares of side xx from a 10 in x 8 in rectangle: V(x)=(10−2x)(8−2x)(x)=80x−36x2+4x3V(x) = (10 - 2x)(8 - 2x)(x) = 80x - 36x^2 + 4x^3
  • Piecewise Functions: Functions that require more than one formula to define the output.
    • General Form: f(x)={formula1amp;if x is in domain 1 formula2amp;if x is in domain 2 formula3amp;if x is in domain 3f(x) = \begin{cases} formula _1 &amp; \text{if } x \text{ is in domain } 1 \ formula _2 &amp; \text{if } x \text{ is in domain } 2 \ formula _3 &amp; \text{if } x \text{ is in domain } 3 \end{cases}
    • Examples:
      • A. Mobile plan with ₱250.00 monthly charge including 200 free texts, and ₱1.00 per excess message: t(m)={250amp;if 0<m≤200 250+mif m>200t(m) = \begin{cases} 250 &amp; \text{if } 0 < m ≤ 200 \ 250 + m & \text{if } m > 200 \end{cases}
      • B. Chocolate bar costs ₱50.00 each, but ₱48.00 each if buying more than 5: f(n)={50namp;if 0<n≤5 48nif n>5f(n) = \begin{cases} 50n &amp; \text{if } 0 < n ≤ 5 \ 48n & \text{if } n > 5 \end{cases}
      • C. Catering service costs ₱250.00 per head for 50 or less, ₱200.00 for 51-100, and ₱150.00 for >100: C(h)={250amp;if n≤50 200amp;if 51≤n≤100 150amp;if ngt;100C(h) = \begin{cases} 250 &amp; \text{if } n ≤ 50 \ 200 &amp; \text{if } 51 ≤ n ≤ 100 \ 150 &amp; \text{if } n &gt; 100 \end{cases}
What’s More
  • Independent Practice 1
    1. A person earns ₱750.00 per day. Express total salary SS as a function of days nn.
      • Answer: S(n)=750nS(n) = 750n.
    2. Jeepney charges ₱9.00 for the first 4 km and ₱0.50 per additional km. Express fare FF as a function of distance dd.
      • Answer: F(d)=9+0.50(d−4)F(d) = 9 + 0.50(d - 4).
  • Independent Assessment 1
    1. Computer shop charges ₱15.00 per hour. Represent rental fee RR as a function of hours tt.
      • Answer: R(t)=15tR(t) = 15t.
    2. Squares of side aa are cut from an 8 in x 6 in rectangle to form a box. Define the volume VV as a function of aa.
      • Answer: V(a)=(8−2a)(6−2a)a=48a−28a2+4a3V(a) = (8 - 2a)(6 - 2a)a = 48a - 28a^2 + 4a^3.
  • Independent Practice 2
    1. Tricycle fare is ₱10.00 for the first 2 km, and ₱8.00 per additional km. Use a piecewise function.
      • Answer: C(d)={10amp;if d≤2 10+8(d−2)amp;if dgt;2C(d) = \begin{cases} 10 &amp; \text{if } d ≤ 2 \ 10 + 8(d - 2) &amp; \text{if } d &gt; 2 \end{cases}
    2. Parking fee is ₱25.00 for the first two hours and ₱5.00 for each additional hour, with a flat rate of ₱100.00 after 12 hours.
      • Answer: p(t)={25amp;if t≤2 25+5(t−2)amp;if 2<t≤12 100if t>12p(t) = \begin{cases} 25 &amp; \text{if } t ≤ 2 \ 25 + 5(t-2) &amp; \text{if } 2 < t ≤ 12 \ 100 & \text{if } t > 12 \end{cases}
  • Independent Assessment 2
    1. Van rental charges ₱5,500.00 for 5 passengers and ₱500.00 per additional passenger.
      • Answer: v(n)={5500amp;if n≤5 5500+500(n−5)amp;if ngt;5v(n) = \begin{cases} 5500 &amp; \text{if } n ≤ 5 \ 5500 + 500(n - 5) &amp; \text{if } n &gt; 5 \end{cases}
    2. Internet costs ₱500.00 for 30 GB, ₱30.00 per excess GB, with a flat rate of ₱1,000.00 for 50 GB and above.
      • Answer: C(g)={500amp;if g≤30 500+30(g−30)amp;if 30lt;glt;50 1000amp;if g≥50C(g) = \begin{cases} 500 &amp; \text{if } g ≤ 30 \ 500 + 30(g - 30) &amp; \text{if } 30 &lt; g &lt; 50 \ 1000 &amp; \text{if } g ≥ 50 \end{cases}
What I Have Learned

A. Analyzing statements about functions:

  1. Incorrect: A relation's first element is the domain, and the second is the range. The statement incorrectly identifies the domain and range.
  2. Incorrect: Function classifications are one-to-one, many-to-one, and onto correspondence.
  3. Correct. In a function machine, inputs are independent variables and outputs are dependent.

B. Significance of functions in real-life situations:

  • Functions provide a mathematical way to represent relationships between quantities in real-world scenarios, offering insights and predictions.

C. When piecewise functions are used:

  • Piecewise functions are used when different formulas or rules apply over different intervals of the input variable, allowing for more accurate modeling of situations with changing conditions.
What I Can Do
  • Identify real-life situations, create sample problems, and write corresponding function equations.
  • Evaluation criteria: relevance to the topic, well-organized presentation, and correct function equations.
Assessment

Multiple Choice:

  1. Which is NOT true about functions?
    • Answer: d. One-to-many correspondence is a function.
  2. What are y values in a relation called?
    • Answer: b. Range
  3. Which table is NOT a function?
    • Answer: a.
  4. What is the domain of the function (table provided)?
    • Answer: c. D: {1, 2, 3, 4, 5}
  5. Which relation(s) is/are function(s)?
    • Answer: c. h = {(6,1), (-2,3), (2, 6), (7, 2)}
  6. Which relation(s) is/are function(s)?
    • Answer: d. the rule which assigns each person a surname
  7. A person encodes 1000 words per hour. Express total words WW as a function of hours nn.
    • Answer: c. W(n)=1000nW(n) = 1000n
  8. Judy earns ₱300.00 per day plus ₱25.00 per hour taking care of a child. Express total salary SS as a function of hours hh.
    • Answer: a. S(h)=300+25hS(h) = 300 + 25h
  9. Which function defines the volume of a cube?
    • Answer: b. V=s3V = s^3, where s is the length of the edge
  10. Eighty meters of fencing encloses a rectangular garden. Give a function AA that represents the area in terms of xx.
    • Answer: a. A(x)=40x−x2A(x) = 40x - x^2
  11. Mobile plan costs ₱400.00 monthly including 500 free texts. Excess messages cost ₱1.00. Represent the monthly cost s(t)s(t).
    • Answer: a. s(t)=400s(t) = 400 if 0 < t ≤ 500
  12. Mobile plan costs ₱400.00 monthly including 500 free texts. Excess messages cost ₱1.00. Represent the monthly cost s(t)s(t).
    • Answer: a. s(t)=400+ts(t) = 400 + t if t > 500
  13. T-shirts cost ₱200.00 each or ₱18,000.00 for 100 shirts and ₱175.00 for each excess. Represent cost t(s)t(s).
    • Answer: d. t(s)=200st(s) = 200s, if 0 < s ≤ 99
  14. T-shirts cost ₱200.00 each or ₱18,000.00 for 100 shirts and ₱175.00 for each excess. Represent cost t(s)t(s).
    • Answer: c. t(s)=18,000t(s) = 18,000, if s=100s = 100
  15. T-shirts cost ₱200.00 each or ₱18,000.00 for 100 shirts and ₱175.00 for each excess. Represent cost t(s)t(s).
    • Answer: a. t(s)=18,000+175(s−100)t(s) = 18,000 + 175(s - 100), if s > 100
Additional Activities
  1. Contaminated water initially has 5 mg/L of pollutants, reduced by 10% each hour. Define the pollutant concentration function.
    • C(t)=5(0.9)tC(t) = 5(0.9)^t
  2. Typhoon rainfall is 25mm/hour for 3 hours, then slows for 1 hour, then 20mm/hour for 2 hours. Write a piecewise function.
    • R(t)={25tamp;if 0≤t≤3 75amp;if 3lt;t≤4 75+20(t−4)amp;if 4lt;t≤6R(t) = \begin{cases} 25t &amp; \text{if } 0 ≤ t ≤ 3 \ 75 &amp; \text{if } 3 &lt; t ≤ 4 \ 75 + 20(t - 4) &amp; \text{if } 4 &lt; t ≤ 6 \end{cases}