Solving Real-Life Problems Involving Functions: A General Mathematics Module, Slides of Mathematics

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General Mathematics

General Mathematics

Quarter 1 – Module 4:

Solving Real-Life Problems

Involving Functions

What I Need to Know

This module was designed and written to help you solve problems involving functions

bearing in mind that you already know how to represent real – life situation using

functions including piece-wise functions, evaluate functions and perform operations

on functions. These skills will aid you in attaining success on this module.

Solving problems involving functions is essential in predicting values that will help

in decision making process. This module covers varied situations that can be seen

in real life such as travel fares, monthly bills sales and the like. It is hoped that upon

exploring this learning kit you will find the eager and enthusiasm in completing the

task required. Best of luck!

After going through this module, you are expected to:

1. represent situations as functions and evaluate functions to determine the

required quantity

1. apply concepts learned in solving real-life problems involving functions
2. solve problems involving functions

What I Know

Choose the letter of the best answer. Write the chosen letter on a separate sheet of

paper.

1. Given ℎ

( 𝑥

)

5 𝑥− 6

3

, determine ℎ( 3 )

a. - 3

b. 3

c. 1

d. - 1

1. Let 𝑓

( 𝑥

) = 3 𝑥

2

− 𝑥 + 5 , find 𝑓(𝑥 + 1 )

a. 3 𝑥

2

• 5 𝑥 + 5

b. 3 𝑥

2

• 5 𝑥 + 7

c. 3 𝑥

2

− 𝑥 + 9

d. 3 𝑥

2

• 5 𝑥 + 9

1. Which of the following expresses the total earnings (E) as a function of the

number ( n ) of days if a laborer works and earning ₱ 400 .00 per day?

a. E (n ) = 400 + n

b. E (n) = 400 ÷ n

c. E (n) = 400(n)

d. E (n ) = 400 – n

1. If the temperature in degrees Celsius inside the Earth is represented by T (d )

= 10 d + 20 where (d ) is the depth in kilometers, what is the temperature inside

the Earth in 10 kilometers?

a. 40 ℃

b. 50 ℃

c. 120 ℃

d. 180 ℃

1. Express the perimeter P of a square with side x as a function of its area

a. 𝐴 =

𝑃

2

16

b. 𝐴 = 16 𝑃

2

c. 𝐴 =

𝑃

2

4

d. 𝐴 =

16

𝑃

2

For numbers 6 – 7 use the problem below:

Cotta National High School has 1,200 students enrolled in 2016 and 1,500 students

in 2019. The student population P grows as a linear function of time (t) , where t is

the number of years after 2016.

1. Which of the following functions represents the student population that

relates to time t?

a. P (t )= 100t + 1, 200

b. P (t) = 1,200t - 100

c. P( t) = 1,200t + 100

d. P (t ) = 100t – 1 , 200

1. How many students will be enrolled in Cotta National High School by 2020?

a. 1, 800

b. 1, 700

c. 1, 600

d. 1, 650

1. How many pages were printed if Mark received a payment of ₱ 600 .00?

a. 100 pages

b. 80 pages

c. 60 pages

d. 50 pages

1. If Mark offers a promo to loyal costumer that the first 20 pages of the printed

output will be free of charge, how much will he charge to a loyal customer who

printed 70 pages of output?

a. ₱ 250.

b. ₱ 50.

c. ₱ 350.

d. ₱ 450.

Lesson

Solving Real-Life Problems

Involving Functions

Majority of the problems we encounter in real life situation involve relationship

between two quantities where one quantity depends on another. For example,

personnel in Department of Health observes the number of persons infected by a

particular virus in a certain community increases with time. In finding out the exact

function relating to the number of persons infected to time, modelling can be used.

Once the model is determined solving and predicting the properties of the subject

being studied can be done. At this point we will focus on solving in order for us to

predict answer to existing problems.

What’s In

YES I CAN!

Listed below are the skills and competencies you should possess before proceeding

to this lesson. Read the statements and assess yourself whether you agree or disagree

with the statements.

Statement Agree Disagree

1. I can carefully read and analyze a given problem
2. I can determine the given and the facts required in

a problem

1. I can represent real – life situation using function,

including piece – wise function

1. I can perform operations on functions
2. I can evaluate functions

• If you agree with all the statements that means you are very much ready with this

module, however if there are some statements where you disagree that means you

need to have a quick review of the previous lesson that will aid you in gaining

success in this lesson.

Let us take a quick tour to what you learn in the previous modules

Solution:

a. 𝑓

( 𝑥

) = 𝑥 + 3 𝑔

( 𝑥

) = 𝑥 − 2

𝑓( 3 ) = 3 + 3 𝑔(− 2 ) = (− 2 ) − 2

𝑓

( 3

) = 6 𝑔

( − 2

) = − 4

𝑓

( 3

)

• 𝑔

( − 2

) = 6 +

( − 4

)

𝑓

( 3

)

• 𝑔

( − 2

) = 2

b. 𝑓

( 𝑥

) = 𝑥 + 3 𝑔

( 𝑥

) = 𝑥 − 2

𝑓( 4 ) = 4 + 3 𝑔( 0 ) = 0 − 2

𝑓

( 4

) = 7 𝑔

( 0

) = − 2

𝑓

( 4

) − 𝑔

( 0

) = 7 − (− 2 )

𝑓

( 4

) − 𝑔

( 0

) = 9

c. 𝑓

( 𝑥

) = 𝑥 + 3 𝑔

( 𝑥

) = 𝑥 − 2

𝑓(𝑥) ∙ 𝑔(𝑥) = (𝑥 + 3 )(𝑥 − 2 )

𝑓(𝑥) ∙ 𝑔(𝑥) = 𝑥

2

• 𝑥 − 6

d. 𝑓

( 𝑥

) = 𝑥 + 3 𝑔

( 𝑥

) = 𝑥 − 2

𝑓

( 9

) = 9 + 3 𝑔

( 8

) = 8 − 2

𝑓

( 9

) = 12 𝑔

( 8

) = 6

𝑓( 9 )

𝑔( 8 )

=

12

6

= 2

At this point you may now proceed to the next section of this module!

Notes to the Teacher

The teacher may also point out the importance of the concept of

zero of linear function in solving problems involving functions. The

zero of a linear function f(x) is the real number a such that f(a)=0.

This suggest that the zero of linear function is found by equating

it to zero and then solving the resulting equation for x. This will be

used in the latter example in the module.

What’s New

JEEPNEY OR TRICYCLE?

Read and analyze the problem below.

Miguel is a senior high school student who commutes from home to school which is

15 km apart. There are two modes of transportation the first one is through jeep and

the other one is through tricycle. In riding a jeepney the fare charge ₱ 9 .00 for the

first 4 km travel and ₱0.75 for each additional kilometer. Meanwhile in riding a

tricycle the fare would be ₱ 10 .00 for the first km travel and ₱ 1 for each additional

kilometer.

Will you help Miguel analyze his situation?

Questions

1. If you are Miguel and decided to ride in a jeepney, how much will be your

fare? _____________________________________________________________

Hint: You may use the table below to compute the fare

No. of km 0 - 4 5 6 7 8 9 10 15

Amount charge 9 9+

9.75+

2. If you decided to ride in a tricycle how much will be your fare? ____________

Hint: You may use the strategy in no. 2

1. What characteristics does Miguel possess if he chose to ride a jeepney?

1. Is there any advantage in riding a jeepney instead of tricycle? Or riding a

tricycle instead of jeepney? What would it be?

1. If you are Miguel which between the two modes of transportation will you

choose? Why? ____________________________________________________

Solution

a. Explore. Since the first step involves analysis and proper labeling of the

known and unknown facts we will let x = number of stations traveled. There

are also some conditions that was set in the problem such as the cost of fare

which is set up to 4 stations only thus we can represent x – 4 = number of

stations traveled over and above 4 stations

Plan. In writing an equation that will represent the relationship between the

known and unknown quantities, since we know that if we travelled up to 4

stations we must pay P18, we can represent it as

f(x) = 18 for 0 < 𝑥 ≤ 4

However, if we travelled more than 4 stations the cost of the fare have different

method of computation so we need to consider it. Since the cost of every

station after the 4

th

station is ₱ 5 .00 we will now obtain

f(x) = 18 + 5( x – 4)

Now simplifying the equation will lead us to:

f(x) = 18 + 5 x – 20

f(x) = 5 x – 2

At this point we can say that the fare function is f(x) = 5x - 2

b. Solve. To find the fare charge for 15 stations the fare function f(x) = 5 x - 2

will be used and 15 will be substituted to the function

f(15) = 5(15) – 2

= 73

By evaluating the function we obtained f(15) = 73

Check. To check whether we arrived at the correct solution you can use

table or graph.

Thus. the proposed fare for 15-station travel is ₱ 73.

c. f(20) = 5 (20) – 2

= 98

The proposed fare for 20 – station travel is ₱ 98

Example no. 2

BINGE WATCH

Lucena Network charges ₱ 450 .00 monthly cable connection fee plus ₱ 130 .00 for

each hour of pay-per-view (PPV) event regardless of a full hour or a fraction of an

hour.

a. Find payment function f(x) where x represents the number of PPV hours.

b. What is the monthly bill of a customer who watched 25 hours of PPV events?

c. What is the monthly bill of a customer who watched 0.5 hour of PPV events?

Solution:

a. ₱ 450 .00 = fixed monthly cable connection fee

Let x = number of PPV hours in a month

₱ 130 .00(x) = amount of PPV payment in a specific hour

The payment function is f(x) = 450 + 130(x).

b. The monthly bill of a customer who watched 25 hours PPV events can be

represented by 24 < x ≤ 25.

f(x) = 450 + 130 (x).

f(25) = 450 + 130(25)

= 450 + 3,

= 3,

The monthly bill of a customer who watched 25 hours of PPV event is ₱3,750. 00

c. The monthly bill of a customer who watched 0.5 hour PPV events can be

represented by 0 < x ≤ 1 and since the problem states that regardless of a full

hour or a fraction of an hour the additional charge will be made on hourly

basis only, thus the value of x will be 1

f(x) = Php 450.00 + Php 130.00 (x).

f(1) = Php 450.00 + Php 130.00(1)

= Php 450.00 + Php 130.

= Php 580.

Independent Practice 2

a. Find the monthly cost function C(x) where x represents the number of

minutes used.

Hint: Monthly Cost = Plan Cost + Additional Charge per Minute

________________________________________________________________

b. How much is the monthly cost incurred if the owner used 180 minutes of

call?____________________________________________________________

c. How much is the monthly cost of the plan if the owner used 300 minutes

of call?______________________________________________________________

Independent Assessment 2

What I Have Learned

According to Alice Hoffman every problem has a solution. In finding the solution one

important aspect to consider is the “how” or the process of finding it. In solving

problems involving functions there are different process that we can employ to attain

the solution.

Hello!

A certain cellphone company offers a plan that costs ₱1,200.00 per

month. The plan includes 180 free minutes of call and charges ₱7.

for each additional minute of usage.

Connected!

CATV Lucena costs ₱1,500.00 a month which also includes 15 GB of

data monthly. It charges ₱50.00 for each additional gigabytes usage.

Find the monthly cost incurred if the owner used 45 GB of data in a

month.

In three to five sentences write the process that you follow in solving problems

involving functions.

What I Can Do

You wanted to join a booth fair, and you are aiming to get a profit that is twice as

your capital. Your starting capital is ₱15,000.00. Make a financial plan for the booth

that you will set up and the product that you will sell. You may use the sample plan

below:

FINANCIAL PLAN

Product: _________________________

Description of product: ________________________

Goal: _____________________________________

Capital: ₱ 1 5,000.

Fixed Cost (Labor, Machineries, Expenses for the booth etc): _______

Variable Cost (Materials, Ingredients, etc): ____________

Profit function: ___________________

Prove that profit function will yield an amount that is twice the capital

1. Which value of x will make Emmanuel’s candy shop suffer loss for selling

packs of chocolates?

a. X> 100

b. x≤ 100

c. x< 100

d. x≥ 100

1. How many chocolate bars must be sold if Emmanuel wanted to earn a profit

of ₱ 6 , 750 .00?

a. 100

b. 150

c. 250

d. 350

1. How much is the gain if Emmanuel sold 350 packs of chocolates?

a. ₱ 6 , 750.

b. ₱ 9 , 750.

c. ₱11. 250.

d. ₱15, 250.

For numbers 6- 10 use the problem below

Mariel wanted to avail a cellphone plan that offers a monthly fee of ₱ 2 ,500.00. It

includes 240 minutes of call and charges ₱7.50 for each additional minute of usage.

1. Which of the following pertains to the monthly cost function?

a. C(x) = 2, 500 - 1800x

b. C(x) = 2,500 + 1800x

c. C(x) = 700 + 7.50x

d. C(x) = 700 – 7.50x

1. What value of x will not require any additional charge in her monthly bill?

a. x > 240

b. x < 240

c. x ≥ 240

d. x ≤ 240

1. How many additional minutes of call did she make, if she paid ₱ 2 , 800 .00 in

her monthly bill?

a. 20 minutes

b. 40 minutes

c. 60 minutes

d. 80 minutes

1. How much is her monthly cost incurred if she made an additional usage of 20

minutes of call?

a. ₱ 2 , 500.

b. ₱ 2 , 600.

c. ₱ 2 , 650.

d. ₱ 3 , 350.

1. How much will she need to pay from using a total of 350 minutes of call in

one month?

a. ₱ 950.

b. ₱ 2 , 610.

c. ₱ 3 , 325.

d. ₱ 4 , 325.

For numbers 11- 15 use the problem below

A local cable network charges ₱ 950 .00 monthly connection fee plus ₱ 100 .00 for

each hour of pay-per-view (PPV) event regardless of a full hour or a fraction of an

hour.

1. Which of the following pertains to the payment function suggested in the

problem?

a. f(x) = 100x + 950

b. f(x) = 100x – 950

c. f(x) = 950x + 100

d. f(x) = 950x – 100

1. What is the monthly bill of a customer who watched 20 hours of PPV events?

a. ₱2, 950.

b. ₱3, 950.

c. ₱4, 950.

d. ₱ 5 , 950.

1. How much is the monthly bill of a customer who watched 0.5 hours of PPV

events?

a. ₱ 950.

b. ₱ 1 , 050.

c. ₱ 2 , 050.

d. ₱ 3 , 050.

1. What will be the monthly bill of a customer who watched 12.3 hours of PPV

events?

a. ₱ 1 , 250.

b. ₱ 2 , 250.

c. ₱ 3 , 250.

d. ₱ 4 , 250.