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General Mathematics

General Mathematics

Quarter 1 – Module 15: Solving

Real-life Problems Involving

Inverse Functions

What I Need to Know

This module was intended and written to guide and help you apply inverse

functions to real-life situations such as finding the original number, conversion

of currency, converting units of temperature from degree Celsius to degree

Farenheit and a lot more.

Likewise, you will learn how to evaluate inverse functions and interpret results.

The knowledge and skills you have learned from the previous lessons are

significant for you to solve real-life problems involving inverse functions.

After going through this module, you are expected to:

1. recall how to finding the inverse of the functions;
2. solve problems involving inverse functions; and

3. evaluate inverse functions and interpret results.

What I Know

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

of paper.

1. Which of the following is the inverse 𝑓

2

a. f

• 1

(x) =

𝑥− 5

3

b. f

• 1

(x) =

𝑥+ 5

3

c. f

• 1

(x) =

𝑥− 3

5

d. f

• 1

(x) =

𝑥+ 3

5

2. Which of the following is the inverse 𝑓

a. 𝑓

− 1

𝑥+ 5

2

b. 𝑓

− 1

𝑥+ 5

2

c. 𝑓

− 1

𝑥− 2

5

d. 𝑓

− 1

𝑥+ 2

5

3. A study found that the relationship between the students’ exam scores (x) and

the number of hours they spent in studying 𝑓(𝑥) is given by the equation of

function 𝑓(𝑥) =

𝑥 − 55

10

. Using this information, what will be the estimated number

of hours that the student spent studying if he scored 85 on the test?

a. 4 hours b. 3 hours c. 2 hours d. 1 hour

4. The relationship between temperatures in degree Fahrenheit (°F) and in degree

Celsius (°C) is given by °𝐹 =

9

5

°𝐶 + 32. What is the corresponding value in degree

Celsius of 100°𝐹?

a. 37.78 °𝐶 b. 42.50°𝐶 c. 65.28°𝐶 d. 89.92°𝐶

For items number 5-7, refer to the following:

Audrey and her mother are planning for a debut party. Audrey suggested that she

wants to celebrate her birthday at Jardin De Emilia Hall. The reception hall costs

a flat rate of ₱2000.00 and an additional rental fee of ₱50.00 per guest. If their

budget for hall expenses is limited at ₱10,00.

5. Which of the following is the cost as a function of the number of guests?

a. y = 2000 + 50x b. y = 2000 – 50x c. y = 50 + 2000x d. y = 50 – 2000x

6. Which of the following is the inverse of cost as a function of the number of

guests?

a. 𝑓

− 1

𝑥 − 50

2000

c. 𝑓

− 1

𝑥 – 2000

50

b. 𝑓

− 1

𝑥+ 50

2000

d.𝑓

− 1

𝑥 + 2000

50

7. What is the domain and range of the inverse?

a. D = {x 𝜖 N | 0 ≤ x ≤ 260} c. D = {x 𝜖 N | 0 ≤ x ≤ 160}

R = {y 𝜖 R | 0 ≤ y ≤ 10,000} R = {y 𝜖 R | 0 ≤ y ≤ 10,000}

b. D = {x 𝜖 N | 0 ≤ x ≤ 2000} d. D = {x 𝜖 N | 0 ≤ x ≤ 10,000}

R = {y 𝜖 R | 0 ≤ y ≤ 10,000} R = {y 𝜖 R | 0 ≤ y ≤ 2000}

8. Suppose I am travelling at 50 miles per hour, and I want to know how I have

gone in x hours. Then, it can be represented by the function 𝑓(𝑥) = 50 𝑥. Find

the inverse of the function.

a. f

• 1

(x) =

𝑥

25

b. f

• 1

(x) =

𝑥

50

c. f

• 1

(x) =

𝑥

75

d. f

• 1

(x) =

𝑥

100

For items number 9-10, refer to the following:

Luis is standing on the ground to take a series of photographs of a kite rising

vertically. The distance between Luis at (B) and the launching point of the kite (A)

is 500 meters. Luis must keep the kite on sight and therefore its angle of elevation

must change with height x of the kite.

9. Find the angle t as a function of the height x.

a. t = tan

• 1

(

𝑥

500 𝑥

) c. t = tan

• 1

(

𝑥

500

b t = tan

• 1

(

500 𝑥

300

d. t =tan

• 1

10. Find the angle t in degrees when x is equal to 150 meters.

a. 25.6 b. 26.6 c. 27 d. 28

11. Find the angle t in degrees when x is equal to 300 meters.

a. 48 b. 47 c. 46 d. 45

Lesson

Solving Real-life Problems

Involving Inverse

Functions

You have learned from your previous modules the representations inverse

functions through its table of values, graphs, and equations. You also learned how

to find its domain and range which are important in the study of solving real-life

problems involving inverse functions. This module will help you solve real-life

problems involving inverse functions.

Let us start your journey by recalling the previous lessons you already

learned about inverse functions. Here is the list of functions and its inverse,

match column B to column A by finding the inverse of the items in column A.

Write the letter of the answer in the box below that will reveal a “word” or the

name of the “building” that you are looking for.

The United Arab Emirates was given the title of “The Tallest Building in

the World” on January 4, 2010. What is the name of the building?

Column A Column B

1. g(x) = x

5

• 3 R. y =

5 𝑥+ 1

9 𝑥− 4

2. f(x) = 7x + 10 A. y =

𝑡

3

3. h(x) =

4 𝑥+ 1

9 𝑥− 5

A. y = ±

𝑥

2

5

4. k =

5

9

(𝑡 − 32 ) + 273. 15 J. y =

9

5

5. w(x) = 2x + 9 L. y = √

𝑥+ 7

2

3

6. t(x) =

𝑥+ 2

3 𝑥− 5

K_. y_ =

𝑥− 9

2

7. r(x) = |5x| B. y = √

5

8. s(x) = 2x

3

• 7 H. y =

5 𝑥+ 2

3 𝑥− 1

9. q(x) = 3x - 5 F. y =

𝑥− 11

5

10. n(x) = 5x + 11 I. y =

𝑥+ 5

3

11. z(x) = 3t R. y = √

𝑥− 2

7

3

U. y =

𝑥− 10

7

What’s In

Notes to the Teacher

To be able to arrive in an accurate and similar answer, the teacher

must advise the learners to recall the steps in finding the inverse of

the function and the properties of an inverse function.

Now, that you already know how to find the inverse of the function, and

how to evaluate inverse functions, as well as finding the domain and range. I am

confident that you are now ready for the new lesson.

Exchange Rate!

Anna’s mother works in South Carolina USA as a domestic helper for a living. She

sends off money in the Philippines each month. Recently the exchange was $ 1.

to ₱50.85.

(a) Complete the table by converting U.S. dollar to peso

(b) Describe how did you convert US dollar to peso.

_____________________________________________________________________

_____________________________________________________________________

(c) Write an equation that converts dollar into peso.

_____________________________________________________________________

_____________________________________________________________________

(d) Write an equation that converts peso into dollar using the equation in

(c).

_____________________________________________________________________

_____________________________________________________________________

What’s New

Steps in finding the inverse of a function is given below.

To find 𝑓

− 1

1. Replace 𝑓(𝑥) with 𝑦.
2. Interchange 𝑥 and 𝑦.
3. Solve for the new y from the equation in Step 2.
4. Replace the new 𝑦 with 𝑓

− 1

(𝑥) if the inverse is a function

For better understanding, study the examples below and reflect on the different

steps to solve real-life problems involving inverse function.

Example 1

Andreau and his friend are playing a number - guessing game. Andreau asks his

friend to think a positive number, then add four to the number. Next, square the

resulting number, and multiply the result by 3. Finally, divide the result by 2. If

you are his friend and you get a result of 50, (a) write an inverse function that will

give you the original number and (b) determine the original number.

Solutions:

To find the inverse, you need first to represent a model for the situation

Let 𝑥 be the number that you think of

𝑥 + 4 represents the statement “add four to the number”

2

represents the statement “square the resulting number”

2

represents the statement “multiply the result by 3”

3 (𝑥 + 4 )

2

2

represents the statement “divide the result by 2”

Therefore, the model for the situation is f(x) =

3 (𝑥 + 4 )

2

2

To find the inverse.

y =

3 (𝑥 + 4 )

2

2

Write 𝑓(𝑥) as y

x =

3 (𝑦+ 4 )

2

2

Interchange x and y

2x = 4(y + 3)

2

Multiply both sides by 2

2 𝑥

4

= (y + 3)

2

Multiply both sides by

1

4

2 𝑥

4

2

Get the square root of both sides

2 𝑥

4

= y + 3

𝑥

2

• 3 = y Apply the addition property of equality

Therefore, the inverse of the function is 𝑓

− 1

𝑥

2

(b) To find the original number, use the inverse of the function 𝑓

− 1

𝑥

2

and evaluate 𝑓

− 1

f

• 1

50

2

f

• 1

(50) = √ 25 – 3

f

• 1

f

• 1

Therefore, the original number is 2.

Example 2

To convert from degrees Fahrenheit to Kelvin, the function is

k(t)=

5

9

(t – 32) + 273.15 , where t is the temperature in Fahrenheit (Kelvin is the SI

unit of temperature). Find the inverse function converting the temperature in

Kelvin to degrees Fahrenheit

Solution:

The equation of the function is: k=

5

9

(t – 32) + 273.

We do not interchange the variables 𝑘 and 𝑡 because it refers to the

temperatures in Kelvin and Fahrenheit respectively.

Solve for t in terms of k:

Use the given formula

k=

5

9

(t – 32) + 273.

k – 273.15 =

5

9

(t – 32) Apply the addition property of equality

9

5

k – 273.15 =

5

9

(t – 32)

9

5

Multiply both sides by

9

5

9

5

) k – 273.15 = (t – 32)

9

5

(k – 273.15)+ 32 = t Apply the addition property of equality

Therefore, the inverse function is t(k)=

9

5

(k – 273.15) + 32 where k is the

temperature in Kelvin

Example 3

The SSG officers of Camohaguin National High School are planning for a JS

Prom. The allocated budget for decorations, sounds, and other miscellaneous

expenses is ₱10,000.00 and an additional ₱150.00 for meal expenses for each

guest. The organization received an amount of ₱40,000.00 from its external

stakeholders.

a. Write the total allocated budget as a function of the number of guests.

b. Find the inverse of the function.

c. State the domain and range for this situation.

d. Find the possible number of guest for a budget of ₱40,000.

Solutions:

(a) Let 𝑥 be number of guest

(b) 𝑥 is the total monthly cost of the service, and 𝐶

− 1

(𝑥) is the number of

songs downloaded.

(c) 15 songs downloaded if a member’s monthly bill is ₱ 3 ,813.

Example 5

Maria wants to buy a particular breed of bangus. And she is aware that the

weight W (in kilograms) of a particular breed of bangus is related to its length L

(in centimeter). Given this function 𝑊 = ( 5. 32 𝑥 10

− 3

2

, find its inverse and

determine the approximate length of a bangus that weighs 0.769 kilogram

Solutions:

(a) To find the inverse

− 3

2

𝑊

5. 32 𝑥 10

− 3

( 5. 32 𝑥 10

− 3

)

5. 32 𝑥 10

− 3

2

Divide both sides by ( 5. 32 𝑥 10

− 3

𝑊

5. 32 𝑥 10

− 3

= L

2

𝑊

5. 32 𝑥 10

− 3

2

Get the square root of both sides.

𝑊

5. 32 𝑥 10

− 3

= L

Therefore, the inverse of the function is L = √

𝑊

5. 32 𝑥 10

− 3

(b) To determine the approximate length of a bangus that weighs 0.

kilogram , evaluate the inverse f

- 1

(L) =

𝑊

5. 32 𝑥 10

− 3

when W=0.769 kilograms

L = √

𝑊

5. 32 𝑥 10

− 3

L = √

1. 769

2. 32 𝑥 10

− 3

L ≈12.

Therefore, the length of a particular breed of bangus is approximately equal to

12.02 cm.

Example 6

The balloon is rising vertically and Dennis wants to take a series of photographs.

The distance between Dennis at (B) and the launching point of the balloon (A) is

250 meters. The angle of elevation must change with the height x of the balloon.

(a) Find the angle t as a function of the height x

(b) Find the angle t in degrees when x is equal to 125, 250, 500 and 1000

meters (approximate your answer to 1 decimal place)

(c) Graph t as a function of x.

Solutions:

(a) The opposite and adjacent sides to angle t are x and 250 meters.

tan ( t ) =

𝑥

250

Use the property of the tangent function and it’s inverse.

tan

• 1

(tan( t )) = x

Rewrite the equation tan ( t ) =

𝑥

250

tan

• 1

(tan( t )) = tan

• 1

𝑥

250

Simplify the left side of the equation to obtain t = tan

• 1

(

𝑥

250

tan

• 1

(tan( t )) = tan

• 1

(

𝑥

250

1

𝑡𝑎𝑛

(tan(t)) = tan

• 1

𝑥

250

𝑡𝑎𝑛𝑡𝑎𝑛 (𝑡)

𝑡𝑎𝑛

= tan

• 1

(

𝑥

250

t = tan

- 1

(

𝑥

250

Therefore, the angle t as a function of the height x is t = tan

- 1

𝑥

250

A

B

What’s More

Read each situation carefully to solve each problem. Write your answer on a

separate sheet of paper.

Activity 1.

(a) Complete the table by converting U.S. dollar to Peso

(b) Describe how did you convert US dollars to Peso.

_____________________________________________________________________

_____________________________________________________________________

(c) Find the inverse of the function to determine the value of a United

States dollar in terms of Philippine Peso on March 13, 2020.

_____________________________________________________________________

_____________________________________________________________________

(d) Interpret and evaluate P (1000) and P

• 1

(1000).

_____________________________________________________________________

_________________________________________________________

Activity 1.

(a) Find the inverse of the function.

________________________________________________________________

________________________________________________________________

(b) How many laptops will produce if the cost is ₱12,000.00?

________________________________________________________________

________________________________________________________________

The ABS CBN News reports foreign exchange rate are closed on March 13,

2020 at ₱51.25. Therefore the formula that gives Philippine Peso in terms of

US dollars on that day is:

P = 51.25D

Where D represents US dollar and P represents Philippine Peso.

The cost of producing laptops by a JOB Company is given by C(x) = 1300x +

5500 (in pesos) where x is the number of produced laptops.

Activity 1.

(a) Write the inverse of the function which converts temperature from

degree Celsius to degree Fahrenheit.

_________________________________________________________________

_________________________________________________________________

(b) Find the equivalent temperatures in degree Fahrenheit of the following

20 °𝐶, 10°𝐶, 5°𝐶, and 0°𝐶.

_________________________________________________________________

_________________________________________________________________

(c) Graph the inverse function.

Activity 1.

(a) Find the inverse of the area in terms of radius.

__________________________________________________________________

__________________________________________________________________

(b) Use the inverse to find the radius of a circle with an area of 48 cm

2

.

__________________________________________________________________

__________________________________________________________________

Activity 1.

a. How far should the supports be if the bridge is to support 6.5 tons?

___________________________________________________________________

___________________________________________________________________

b. Construct an inverse function to determine the result.

___________________________________________________________________

___________________________________________________________________

The formula for converting Celsius to Fahrenheit is given by 𝐹 =

9

5

𝐶 + 32 where C

is the temperature in degree Celsius and F is the temperature in degree Fahrenheit.

Juan is making a collage, and he planned to form a circle by putting together various

pieces of construction paper. Given the formula of the area of the circle 𝐴 = 𝜋𝑟

2

Engineers have determined that the maximum force t in tons that a particular

bridge can carry is related the distance d in meters between its supports by the

following function: 𝑡(𝑑) =

12. 5

𝑑

3

Your output will be graded using this rubric.

CRITERIA EXCELLENT

4 points

SATISFACTORY

3 points

DEVELOPING

2 points

BEGINNING

1point

Accuracy of the

Solution

Shows accurate

solution and

estimation of

the possible

expenses.

Shows solution and

estimation of the

possible expenses

with minimal

errors.

Shows solution

and estimation

of the possible

expenses with

plenty of errors.

The solution and

estimation of the

possible expenses

are all erroneous.

Mathematical

Concept

Shows excellent

understanding

of the concept

of solving real-

life problems

involving

inverse

functions and

other concepts

related to the

problem.

Shows clear

understanding of

the concept of

solving real-life

problems involving

inverse functions.

Shows limited

understanding of

the concept of

solving real-life

problems

involving inverse

functions.

Did not apply the

concept of solving

real-life problems

involving inverse

functions.

Assessment

Multiple Choice. Choose the letter of the best answer. Write the chosen letter on a

separate sheet of paper.

1. Which of the following is the inverse 𝑓(𝑥) = √

𝑥+ 3

7

a. f

• 1

(x) = 7x

2

• 3 b. f

• 1

(x) = 7x

2

• 3 c. f
  • 1

(x) = 3x

2

• 7 d. f

• 1

(x) = 3x

2

• 7

2. Which of the following is the inverse 𝑓

a. 𝑓

− 1

𝑥− 5

6

b. 𝑓

− 1

𝑥+ 5

6

c. 𝑓

− 1

𝑥− 2

5

d. 𝑓

− 1

𝑥+ 2

5

3. A study found that the relationship between the number of hours (x) and the

student’s exam scores 𝑓(𝑥) is given by the equation of function 𝑓

Using this information, what will be the estimated number of scores of the

student if he spent 4 hours in studying?

a. 95 b. 85 c. 75 d. 65

4. The relationship between temperatures in degree Celsius (°C) and in degree

Fahrenheit (°F) is given by °𝐶 =

5

9

(°𝐹 − 32 ). What is the corresponding value in

degree Fahrenheit of 37 .78°𝐶?

a. 80°𝐹 b. 90°𝐹 c. 100°𝐹 d. 110°𝐹

For items number 5-7, refer to the following:

Cath and Arvin are planning for their wedding. Cath suggested that she wants

Casa de Aurora to cater their reception. The reception hall rental fee starts at a

flat rate of ₱ 3 ,500.00 and an additional rental fee of ₱60.00 per guest. If their

budgetis limited at ₱20,000.00.

5. Which of the following represents the total rental fee as a function of the

number of guests?

a. y = 3500 + 60x c. y = 60 + 3500x

b. y = 3500 – 60x d. y = 60 – 3500x

6. Which of the following is the inverse function in item 5?

a. 𝑓

− 1

𝑥 − 60

2000

c. 𝑓

− 1

𝑥+ 35000

50

b. 𝑓

− 1

𝑥+ 50

3500

d. 𝑓

− 1

𝑥− 3500

60

7. What is the domain and range of the inverse?

a. D = {x 𝜖 N | 0 ≤ x ≤ 275} c. D = {x 𝜖 N | 0 ≤ x ≤ 160}

R = {y 𝜖 R | 0 ≤ y ≤ 20,000} R = {y 𝜖 R | 0 ≤ y ≤ 20,000}

b. D = {x 𝜖 N | 0 ≤ x ≤ 2000} d. D = {x 𝜖 N | 0 ≤ x ≤ 10,000}

R = {y 𝜖 R | 0 ≤ y ≤ 10,000} R = {y 𝜖 R | 0 ≤ y ≤ 2000}

8. Suppose I am travelling at 30 miles per hour, and I want to know how I have

gone in x hours. Then, this could be represented by the function 𝑓(𝑥) = 30 𝑥.

Find the inverse of the function.

a. f

• 1

(x) =

𝑥

10

b. f

• 1

(x) =

𝑥

20

c. f

• 1

(x) =

𝑥

30

d. f

• 1

(x) =

𝑥

100

For items number 9-11, refer to the following:

Marx is standing on the ground to take a series of photographs of a kite rising

vertically. The distance between Luis at (B) and the launching point of the kite (A)

is 800 meters. Luis must keep the kite on sight and therefore its angle of elevation

must change with height x of the kite.

9. Find the angle t as a function of the height x.

a. t = tan

• 1

(

𝑥

800

) c. t = tan

• 1

(

800 𝑥

500

b. t = tan

• 1

(

500 𝑥

300

d. t =tan

• 1

10. Find the angle t in degrees when x is equal to 150 meters

a. 31.6 b. 21.6 c. 11.6 d. 10.

11. Find the angle t in degrees when x is equal to 300 meters.

a. 20.6 b. 21.6 c. 22.6 d. 23.

For items number 12-13, refer to the following:

The function defined by 𝑔(𝑥) =

𝑥

5. 3

converts a volume of x liters into g(x) gallons.

12. Which of the following is the inverse of 𝑔(𝑥)?

a. g

• 1

(x) =5.3x c. g

• 1

(x) =

5. 3 𝑥

6. 3 +𝑥

b. g

• 1

(x) =

𝑥

5 𝑥+ 3

d. g

• 1

(x) =

3 𝑥

5. 3