General Mathematics Quarter 1 Module 3: Operations on Functions

General Mathematics Quarter 1 – Module 3: Operations on Functions

Senior High School

General Mathematics

Quarter 1 – Module 3: Operations on Functions

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CO_Q1_General Mathematic SHS Module 3

General Mathematics Alternative Delivery Mode Quarter 1 – Module 3: Operations on Functions First Edition, 2021

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Development Team of the Module

Writers: Nestor N. Sandoval
Editors: Elizabeth B. Dizon, Anicia J. Villaruel, Roy O. Natividad
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Senior High School

General Mathematic

Quarter 1 – Module 3: Operations on Functions

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

What I Need to Know

In this module, the different operations on functions were discussed. Examples were provided for you to be able to learn the five (5) operations: addition, subtraction, multiplication, division and composition of functions. Aside from algebraic solutions, these examples were illustrated, represented in tables and/or mapping diagram for better understanding of the concepts. Activities were provided to enhance your learning. Finally, your task is to answer a guided real-world problem that involves operations on functions.

After going through this module, you are expected to:

1. define operations on functions;
2. identify the different operations on functions; and
3. perform addition, subtraction, multiplication, division, and composition of functions.

What I Know

Write the letter of the correct answer on a separate sheet of paper.

1. The statement “𝑝𝑝(𝑥𝑥) − 𝑞𝑞(𝑥𝑥) is the same as 𝑞𝑞(𝑥𝑥) − 𝑝𝑝(𝑥𝑥)”, 𝑝𝑝(𝑥𝑥) ≠ 𝑞𝑞(𝑥𝑥) is _____.

   • a. always true
   • b. never true
   • c. sometimes true
   • d. invalid

2. Given ℎ(𝑥𝑥) = 2𝑥𝑥² − 7𝑥𝑥 and 𝑟𝑟(𝑥𝑥) = 𝑥𝑥² + 𝑥𝑥 − 1, find (ℎ + 𝑟𝑟)(𝑥𝑥).

   • a. 2𝑥𝑥² – 1
   • b. 3𝑥𝑥² + 6𝑥𝑥 – 1
   • c. 3𝑥𝑥⁴ − 6𝑥𝑥² – 1
   • d. 3𝑥𝑥² − 6𝑥𝑥 – 1

3. Given: 𝑓𝑓(𝑎𝑎) = 2𝑎𝑎 + 1 and 𝑔𝑔(𝑎𝑎) = 3𝑎𝑎 − 3. Find 𝑓𝑓(𝑎𝑎) + 𝑔𝑔(𝑎𝑎)

   • a. 5𝑎𝑎 − 2
   • b. −5𝑎𝑎 + 2
   • c. −2𝑎𝑎 + 1
   • d. −6𝑎𝑎 − 1

4. 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥 − 4 and ℎ(𝑥𝑥) = 2𝑥𝑥 − 7. Find (𝑔𝑔 + ℎ)(3).

   • a. -7
   • b. 1
   • c. -1
   • d. 8

5. 𝑓𝑓(𝑥𝑥) = 6𝑥𝑥² + 7𝑥𝑥 + 2 and 𝑔𝑔(𝑥𝑥) = 5𝑥𝑥² − 𝑥𝑥 − 1, find (𝑓𝑓 − 𝑔𝑔)(𝑥𝑥).

   • a. 𝑥𝑥² + 8𝑥𝑥 + 3
   • b. 5𝑥𝑥² + 8𝑥𝑥 – 1
   • c. 𝑥𝑥² + 6𝑥𝑥 – 1
   • d. 𝑥𝑥² + 8𝑥𝑥 − 1

6. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 − 8 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 + 3. Find 𝑓𝑓(𝑥𝑥) • 𝑔𝑔(𝑥𝑥)

   • a. 𝑥𝑥² + 24
   • b. 𝑥𝑥² − 5𝑥𝑥 + 24
   • c. 𝑥𝑥² − 5𝑥𝑥 − 24
   • d. 𝑥𝑥² + 5𝑥𝑥 + 24

7. If 𝑝𝑝(𝑥𝑥) = 𝑥𝑥 − 1 and 𝑞𝑞(𝑥𝑥) = 𝑥𝑥 − 1, what is 𝑝𝑝(𝑥𝑥) • 𝑞𝑞(𝑥𝑥)

   • a. 𝑥𝑥² + 1
   • b. 𝑥𝑥² + 2𝑥𝑥 − 1
   • c. 𝑥𝑥² − 2𝑥𝑥 + 1
   • d. 𝑥𝑥² − 1

8. Given ℎ(𝑥𝑥) = 𝑥𝑥 − 6 and 𝑠𝑠(𝑥𝑥) = 𝑥𝑥² − 13𝑥𝑥 + 42. Find (ℎ/𝑠𝑠)(𝑥𝑥).

   • a. 1/(𝑥𝑥−7)
   • b. 𝑥𝑥 − 7
   • c. (𝑥𝑥−6)/(𝑥𝑥−7)
   • d. 𝑥𝑥 − 6

9. 𝑔𝑔(𝑥𝑥) = 6𝑥𝑥 − 7 and ℎ(𝑥𝑥) = 5𝑥𝑥 − 1. Find 𝑔𝑔(ℎ(𝑥𝑥))

   • a. −9𝑥𝑥 + 11
   • b. 9𝑥𝑥² + 4𝑥𝑥
   • c. 30𝑥𝑥 + 13
   • d. 30𝑥𝑥 − 13

10. If 𝑗𝑗(𝑥𝑥) = √(𝑥𝑥 + 6) and 𝑘𝑘(𝑥𝑥) = 9 − 𝑥𝑥. Find 𝑗𝑗(𝑘𝑘(−1))

    • a. 9 − √5
    • b. √14
    • c. 16
    • d. 4

For numbers 11-13, refer to figure below

[chart: mapping diagram; figure for numbers 11-13, with 𝑝𝑝(𝑥𝑥) = (𝑥𝑥 + 3)/(𝑥𝑥 − 1) and 𝑞𝑞(𝑥𝑥) = 2𝑥𝑥 + 1]

Values read from the arrows of the diagram:

Input to p(x)	Output of p(x) (input to q(x))	Output of q(x)
-3	0	1
-1	-1	-1
0	-3	-5
2	5	11
3	3	7
5	2	5

11. Evaluate 𝑝𝑝(5)

    • a. 0
    • b. 3
    • c. 2
    • d. 7

12. Find 𝑞𝑞(𝑝𝑝(0))

    • a. -3
    • b. 1
    • c. -3
    • d. -5

13. Find (𝑞𝑞 ∘ 𝑝𝑝)(3)

    • a. 3
    • b. 5
    • c. 7
    • d. -1

For numbers 14-15, refer to the table of values below

[chart: table of values; 𝑚𝑚(𝑥𝑥) = 3𝑥𝑥 − 5 and 𝑛𝑛(𝑥𝑥) = 𝑥𝑥² − 2𝑥𝑥 + 1]

𝑥𝑥	0	1	2	3	4	5	6	7	8
𝑚𝑚(𝑥𝑥)	-5	-2	1	4	7	10	13	16	19
𝑛𝑛(𝑥𝑥)	1	0	1	4	9	16	25	36	49

14. Find (𝑚𝑚/𝑛𝑛)(7)

    • a. 4/9
    • b. 9/4
    • c. 1
    • d. 0

15. Find (𝑛𝑛 ∘ 𝑚𝑚)(4)

    • a. 9
    • b. 16
    • c. 19
    • d. 36

Lesson 1 Operations on Functions

Operations on functions are similar to operations on numbers. Adding, subtracting and multiplying two or more functions together will result in another function. Dividing two functions together will also result in another function if the denominator or divisor is not the zero function. Lastly, composing two or more functions will also produce another function.

The following are prerequisite skills before moving through this module:

• Rules for adding, subtracting, multiplying and dividing fractions and algebraic expressions, real numbers (especially fractions and integers).
• Evaluating a function.

A short activity was provided here for you to help in recalling these competencies. If you feel that you are able to perform those, you may skip the activity below. Enjoy!

What’s In

SECRET MESSAGE

Answer each question by matching column A with column B. Write the letter of the correct answer at the blank before each number. Decode the secret message below using the letters of the answers.

Column A	Column B
_____1. Find the LCD of 1/3 and 2/7.	A. (x + 4)(x − 3)
_____2. Find the LCD of 3/(x−2) and 1/(x+3).	C. (4x+7)/(x²+x−6)
_____3. Find the sum of 1/3 and 2/7.	D. ((x−3)(x+5))/((x−6)(x+3))
_____4. Find the sum of 2/x + 5/x.	E. (x − 2)(x + 3) or x² + x − 6
_____5. Find the product of 3/8 and 12/5.	G. (x+4)/(x+2)
_____6. Find the sum of 3/(x−2) and 1/(x+3).	H. (x + 1)(x − 6)

For numbers 7-14, find the factors.

Column A	Column B
_____7. x² + x − 12	I. 13/21
_____8. x² − 5x − 6	L. (𝑥𝑥 − 4(𝑥𝑥 − 3)

Column A	Column B
_____9. x² + 6x + 5	M. −5
_____10. x² + 7x + 12	N. 21
_____11. x² − 7x + 12	O. (𝑥𝑥 − 5)(𝑥𝑥 − 3)
_____12. x² − 5x − 14	R. (x + 4)(x + 3)
_____13. x² − 8x + 15	S. (𝑥𝑥 − 7)(𝑥𝑥 − 5)
_____14. x² − 12x + 35	T. 9/10
_____15. Find the product of (x²+x−12)/(x²−5x−6) and (x²+6x+5)/(x²+7x+12).	U. (𝑥𝑥 − 7(𝑥𝑥 + 2)
_____16. Divide (x²+x−12)/(x²−5x−14) by (x²−8x+15)/(x²−12x+35).	W. 7/𝑥𝑥
_____17. In the function f(x) = 4 − x², find f(−3).	Y. (x + 5)(x + 1)

Secret Message:

---

4 2 11 6 13 17 2 5 13 14 2 1 3 13 10

---

8 3 16 8 14 6 8 13 13 11 3 17

---

16 11 7 15 9 13 12 7 10 2 8 2 10 2

What’s New

SAVE FOR A CAUSE

Thru inspiration instilled by their parents and realization brought by Covid-19 pandemic experience, Neah and Neoh, both Senior High School students decided to save money for a charity cause. Neah has a piggy bank with ₱10.00 initial coins inside. She then decided to save ₱5.00 daily out of her allowance. Meanwhile, Neoh who also has a piggy bank with ₱5.00 initial coin inside decided to save ₱3.00 daily. Given the above situation, answer the following questions:

a. How much money will be saved by Neah and Neah after 30 days? after 365 days or 1 year? their combined savings for one year? b. Is the combined savings enough for a charity donation? Why? c. What values were manifested by the two senior high school students? d. Will you do the same thing these students did? What are the other ways that you can help less fortunate people? e. Do you agree with the statement of Pope John Paul II said that “Nobody is so poor he has nothing to give, and nobody is so rich he has nothing to receive”? Justify your answer. f. What functions can represent the amount of their savings in terms of number of days?

What is It

In the previous modules, you learned to represent real life situations to functions and evaluate a function at a certain value. The scenario presented above is an example of real world problems involving functions. This involves two functions representing the savings of the two senior high school students.

Below is the representation of two functions represented by a piggy bank:

Neah	Neoh	Combined
$f(x) = 5x + 10$	$g(x) = 3x + 5$	$h(x) = 8x + 15$

[illustration: two piggy banks with a plus sign between them and an equals sign leading to a third piggy bank]

Suppose that we combine the piggy banks of the two students, the resulting is another piggy bank. It’s just like adding two functions will result to another function.

Definition. Let f and g be functions.

1. Their sum, denoted by $f + g$, is the function denoted by $(f + g)(x) = f(x) + g(x)$.
2. Their difference, denoted by $f – g$, is the function denoted by $(f – g)(x) = f(x) – g(x)$.
3. Their product, denoted by $f \bullet g$, is the function denoted by $(f \bullet g)(x) = f(x) \bullet g(x)$.
4. Their quotient, denoted by $f/g$, is the function denoted by $(f/g)(x) = f(x)/g(x)$, excluding the values of x where $g(x) = 0$.
5. The composite function denoted by $(f \circ g)(x) = f(g(x))$. The process of obtaining a composite function is called function composition.

Example 1. Given the functions:

$f(x) = x + 5$ $g(x) = 2x – 1$ $h(x) = 2x^2 + 9x – 5$

Determine the following functions:

a. $(f + g)(x)$ b. $(f – g)(x)$ c. $(f \bullet g)(x)$ d. $\left(\frac{h}{g}\right)(x)$ e. $(f + g)(3)$ f. $(f – g)(3)$ g. $(f \bullet g)(3)$ h. $\left(\frac{h}{g}\right)(3)$

Solution:

a. $(f + g)(x) = f(x) + g(x)$ definition of addition of functions $= (x + 5) + (2x – 1)$ replace f(x) and g(x) by the given values $= 3x + 4$ combine like terms

b. $(f – g)(x) = f(x) – g(x)$ definition of subtraction of functions $= (x + 5) – (2x – 1)$ replace f(x) and g(x) by the given values $= x + 5 – 2x + 1$ distribute the negative sign $= -x + 6$ combine like terms

c. $(f \bullet g)(x) = f(x) \bullet g(x)$ definition of multiplication of functions $= (x + 5) \bullet (2x – 1)$ replace f(x) and g(x) by the given values $= 2x^2 + 9x – 5$ multiply the binomials

d. $\left(\frac{h}{g}\right)(x) = \frac{h(x)}{g(x)}$ definition of division of functions $= \frac{2x^2 + 9x – 5}{2x – 1}$ replace h(x) and g(x) by the given values $= \frac{(x + 5)(2x – 1)}{2x – 1}$ factor the numerator $= \frac{(x + 5)\cancel{(2x – 1)}}{\cancel{2x – 1}}$ cancel out common factors $= x + 5$

e. $(f + g)(3) = f(3) + g(3)$ Solve for $f(3)$ and $g(3)$ separately: $f(x) = x + 5$ $g(x) = 2x – 1$ $f(3) = 3 + 5$ $g(3) = 2(3) – 1$ $= 8$ $= 5$

$\therefore f(3) + g(3) = 8 + 5 = 13$

Alternative solution: We know that $(f + g)(3)$ means evaluating the function $(f + g)$ at 3.

$(f + g)(x) = 3x + 4$ resulted function from item a $(f + g)(3) = 3(3) + 4$ replace x by 3 $= 9 + 4$ multiply $= 13$ add

For item $f$ to $h$ we will use the values of $f(3) = 8$ and $g(3) = 5$

f. $(f – g)(3) = f(3) – g(3)$ definition of subtraction of functions $= 8 – 5$ replace f(3) and g(3) by the given values $= 3$ subtract

Alternative solution: $(f – g)(x) = -x + 6$ resulted function from item b $(f – g)(3) = -3 + 6$ replace x by 3 $= 3$ simplify

g. $(f \bullet g)(3) = f(3) \bullet g(3)$ definition of multiplication of functions $= 8 \bullet 5$ replace f(3) and g(3) by the given values $= 40$ multiply

Alternative solution: $(f \bullet g)(x) = 2x^2 + 9x – 5$ resulted function from item c $(f \bullet g)(3) = 2(3)^2 + 9(3) – 5$ replace x by 3 $= 2(9) + 27 – 5$ square and multiply $= 18 + 27 – 5$ multiply $= 40$ simplify

h. $\left(\frac{h}{g}\right)(3) = \frac{h(3)}{g(3)}$

Solve for $h(3)$ and $g(3)$ separately: $h(x) = 2x^2 + 9x – 5$ $g(x) = 2x – 1$ $h(3) = 2(3)^2 + 9(3) – 5$ $g(3) = 2(3) – 1$ $= 18 + 27 – 5$ $= 5$ $= 40$

$\therefore \left(\frac{h}{g}\right)(3) = \frac{h(3)}{g(3)} = \frac{40}{5} = 8$

Alternative solution: $\left(\frac{h}{g}\right)(x) = x + 5$ resulted function from item d $\left(\frac{h}{g}\right)(x) = 3 + 5$ replace x by 3 $= 8$ simplify

Can you follow with what has been discussed from the above examples? Notice that addition, subtraction, multiplication, and division can be both performs on real numbers and functions.

The illustrations below might help you to better understand the concepts on function operations.

In the illustrations, the numbers above are the inputs which are all 3 while below the function machine are the outputs. The first two functions are the functions to be added, subtracted, multiplied and divided while the rightmost function is the resulting function.

Addition Subtraction Multiplication Division

[chart: function machine diagrams; Addition, Subtraction, Multiplication, Division]

Operation	Function machines (input 3)	Output
Addition	f: $x+5$, $f(3)=8$ + g: $2x-1$, $g(3)=5$ = $3x+4$, $(f+g)(3)=13$	13
Subtraction	f: $x+5$, $f(3)=8$ − g: $2x-1$, $g(3)=5$ = $-x+6$, $(f-g)(3)=3$	3
Multiplication	f: $x+5$, $f(3)=8$ • g: $2x-1$, $g(3)=5$ = $2x^2+9x-5$, $(f\bullet g)(3)=40$	40
Division	h: $2x^2+9x-5$, $h(3)=40$ ÷ g: $2x-1$, $g(3)=5$ = $x+5$, $(h/g)(3)=8$	8

Notes to the Teacher

Give emphasis to the students that performing operations on two or more functions results to a new function. The function $(f + g)(x)$ is a new function resulted from adding $f(x)$ and $g(x)$. The new function can now be used to evaluate $(f + g)(3)$ and it will be the same as adding $f(3)$ and $g(3)$.

Composition of functions:

In composition of functions, we will have a lot of substitutions. You learned in previous lesson that to evaluate a function, you will just substitute a certain number in all of the variables in the given function. Similarly, if a function is substituted to all variables in another function, you are performing a composition of functions to create another function. Some authors call this operation as “function of functions”.

Example 2. Given $f(x) = x^2 + 5x + 6$, and $h(x) = x + 2$

Find the following:

a. $(f \circ h)(x)$ b. $(f \circ h)(4)$ c. $(h \circ f)(x)$

Solution.

a. $(f \circ h)(x) = f(h(x))$ definition of function composition $= f(x + 2)$ replace h(x) by x+2

Since $f(x) = x^2 + 5x + 6$ given $f(x + 2) = (x + 2)^2 + 5(x + 2) + 6$ replace x by x+2 $= x^2 + 4x + 4 + 5x + 10 + 6$ perform the operations $= x^2 + 9x + 20$ combine similar terms

Composition of function is putting a function inside another function. See below figure for illustration.

[chart: function composition diagram; h(x) box labeled x+2 feeds into f box labeled x²+5x+6]

Labels printed in the diagram: “Substitute x+2 in all x of x²+5x+6”; “(x+2)²+5(x+2)+6”; “x²+4x+4+5x+10+6”; “(f∘h)(x)”; “x²+9x+20”.

a. $(f \circ h)(4) = f(h(4))$

Step 1. Evaluate h(4) $h(x) = x + 2$ $h(4) = 4 + 2$ $= 6$

Step 2. Evaluate f(6) $f(x) = x^2 + 5x + 6$ $f(6) = 6^2 + 5(6) + 6$ $= 36 + 30 + 6$ $= 72$

$(f \circ h)(4) = f(h(4))$ $= f(6)$ $\therefore = 72$

To evaluate composition of function, always start with the inside function (from right to left). In this case, we first evaluated h(4) and then substituted the resulted value to f(x).

Alternative solution: $(f \circ h)(x) = f(h(x))$ definition of function composition $f(h(x)) = x^2 + 9x + 20$, from item a $(f \circ h)(4) = 4^2 + 9(4) + 20$ replace all x’s by 4 $= 16 + 36 + 20$ perform the indicated operations $= 72$ simplify

[chart: function composition mapping; h(x) = x+2 feeds into f(x) = x²+5x+6; h(4)=6 and f(h(4))=f(6)=72]

A mapping diagram can also help you to visualize the concept of evaluating a function composition.

[chart: mapping diagram; h(x) = x + 2 and f(x) = x² + 5x + 6; left column 1, 4; middle column 3, 6; right column 30, 72 (72 circled)]

From the definition of function composition, $(f \circ h)(4) = f(h(4))$. Looking at the mapping diagram for values and working from right to left, $h(4) = 6$. Substituting 6 to h(4) we have f(6). From the diagram, f(6) is equal to 72. Therefore, $(f \circ h)(4) = f(h(4)) = 72$. In the diagram, the first function h(x) served as the inside function while the second function f(x) is the outside function.

A table of values is another way to represent a function. The mapping diagram above has a corresponding table of values below:

$h(x) = x + 2$ $f(x) = x^2 + 5x + 6$

x	1	3	4	6
h(x)	3	5	6	8
f(x)	12	30	42	72

$(f \circ h)(4) = f(h(4))$ definition of composition of functions $= f(6)$ substitute h(4) by 6 $= 72$ from the table

b. $(h \circ f)(x) = h(f(x))$ definition of composition of functions $= h(x^2 + 5x + 6)$, substitute f(x) by $x^2 + 5x + 6$, given

Since $h(x) = x + 2$ given $h(x^2 + 5x + 6) = x^2 + 5x + 6 + 2$ substitute x by $x^2 + 5x + 6$ $= x^2 + 5x + 8$ combine similar terms

Notes to the Teacher

The functions $(f \circ h)(x)$ and $(h \circ f)(x)$ are generally not the same as we see in the previous examples. It only means that order of functions counts in composition of function operation. There are special cases where they will be the same; this is when the two functions are inverses. Graphing and finding the domain and range of algebraic operations is not covered by this module but this is an interesting activity that can be used as enrichment once this module was mastered.

What’s More

Activity 1: MATCHING FUNCTIONS

Match column A with column B by writing the letter of the correct answer on the blank before each number

Given: $a(x) = x + 2$ $b(x) = 5x – 3$ $c(x) = \frac{x + 5}{x – 7}$ $d(x) = \sqrt{x + 5}$ $e(x) = \frac{3}{x – 7}$

Column A	Column B
_____1. $(a + b)(x)$	a. $\frac{3}{x+5}$
_____2. $(a \bullet b)(x)$	b. $\pm 3$
_____3. $(d \circ a)(x)$	c. $-7$
_____4. $\left(\frac{e}{c}\right)(x)$	d. $-\frac{4}{5}$
_____5. $(c – e)(x)$	e. $\sqrt{x + 7}$
_____6. $(a + b)(-1)$	f. $\frac{x+2}{x-7}$
_____7. $(a \bullet b)(0)$	g. $6x – 1$
_____8. $(d \circ a)(2)$	h. $1$
_____9. $\left(\frac{e}{c}\right)(-2)$	i. $-6$
_____10. $(c – e)(2)$	j. $5x^2 + 7x – 6$

Activity 2: LET’S SIMPLIFY

A. Let $p(x) = 2x^2 + 5x – 3$, $m(x) = 2x – 1$, and $h(x) = \frac{x+1}{x-2}$. Find:

1. $(m – p)(x)$
2. $p(5) + m(3) – h(1)$
3. $\frac{m(x)}{p(x)}$
4. $p(x + 1)$
5. $p(3) – 3(m(2))$

B. Given the following:

• m(x) = 5x − 3
• n(x) = x + 4
• c(x) = 5x² + 17x − 12
• t(x) = (x−5)/(x+2)

Determine the following functions.

1. (m + n)(x)
2. (m · n)(x)
3. (n − c)(x)
4. (c/m)(x)
5. (m ∘ n)(x)
6. (n ∘ c)(−3)
7. n(m(m(2)))

C. Given the functions g(x) = x² − 4 and h(x) = x + 2, Express the following as the sum, difference, product, or quotient of the functions above.

1. p(x) = x − 2
2. r(x) = x² + x − 2
3. s(x) = x³ + 2x² − 4x − 8
4. t(x) = −x² + x + 6

D. Answer the following:

1. Given h(x) = 3x² + 2x − 4, what is h(x − 3)?
2. Given n(x) = x + 5 and p(x) = x² + 3x − 10, find:
   a. (n − p)(x) + 3p(x)
   b. n(x)/p(x)
   c. (p ∘ n)(x)
3. Let m(x) = √(x + 3), n(x) = x³ − 4, and p(x) = 9x − 5, find (m ∘ (n − p))(3).
4. Given w(x) = 3x − 2, v(x) = 2x + 7 and k(x) = −6x − 7, find (w − v − k)(2)
5. If s(x) = 3x − 2 and r(x) = 2/(x+5), find 2(s + r)(x)
6. Given a(x) = 4x + 2, b(x) = (3/2)x, and c(x) = x − 5, find (a • b • c)(x)

What I Have Learned

Complete the worksheet below with what have you learned regarding operations on functions. Write your own definition and steps on performing each functions operation. You may give your own example to better illustrate your point.

Addition	Subtraction	Multiplication	Division	Composition

What I Can Do

Read and understand the situation below, then answer the questions that follow.

The outbreak of coronavirus disease 2019 (COVID-19) has created a global health crisis that has had a deep impact on the way we perceive our world and our everyday lives, (https://www.frontiersin.org).Philippines, one of the high-risk countries of this pandemic has recorded high cases of the disease. As a student, you realize that Mathematics can be a tool to better assess the situation and formulate strategic plan to control the disease.

Suppose that in a certain part of the country, the following data have been recorded.

d	0	1	2	3	4	5	6	7	8
I(d)	3	5	9	12	18	25	35	47	82

Where I(d) is the function of the number of people who got infected in d days

The number of recoveries was also recorded in the following table as the function R(i) where R as the number of recoveries is dependent to number of infected (I).

I	3	5	9	12	18	25	35	47	82
R(I)	0	1	2	5	7	9	12	18	25

a. Evaluate the following and then interpret your answer.

1. R(I(3))
2. R(I(8))
3. I(R(18))

b. The number of deaths (M) was also dependent on the number of infected (I). Complete the table with your own number of deaths values for the given number of infected.

I	3	5	9	12	18	25	35	47	82
M(I)	0	0	1	1	1	2	3	4	6

Evaluate the following and then interpret your answer.

1. M(I(1))
2. M(I(4))
3. I(M(12))

c. What can you conclude about the data presented?

d. What can you suggest to the government to solve the problem?

Assessment

Write the letter of the correct answer on a separate answer sheet.

1. The following are notations for composite functions EXCEPT,
   a. h(p(x)) c. (s ∘ t)(x)
   b. b. f(x)g(x) d. f(g(x))

   
   

2. Find h(3) + d(2) if h(x) = x − 1 and d(x) = 7x + 3
   a. 2 c. 14
   b. b. 5 d. 19

   
   

3. t(x) = −x² + 7x + 1 and r(x) = 5x² − 2x + 8, find (t − r)(2).
   a. 18 c. -13
   b. b. -18 d. 13

   
   

4. f(x) = 4x + 2 and g(x) = 3x − 1, find (f − g)(4).
   a. 0 c. 7
   b. -9 d. -8

   
   

5. If g(x) = x − 4 and f(x) = x + 5, find f(x) • g(x)
   a. x² + x + 20 c. x² − x − 20
   b. x² − x + 20 d. x² + x − 20

   
   

6. Given h(n) = (n+6)/(n−4) and p(k) = (n+6)/(n²+4n−32). Find (h/p)(k).
   a. 1/(n+8) c. 1/(n−8)
   b. b. n − 8 d. n + 8

   
   

7. If f(x) = 18x² and t(x) = 8x, find (f/t)(x).
   a. 9x/4 c. 4/(9x)
   b. b. 4x/9 d. 9/(4x)

   
   

8. When f(x) = 3x − 5 and g(x) = 2x² − 5, find f(g(x)).
   a. x² + 2x + 3 c. 6x² + 20
   b. b. 6x² − 20 d. 2x² + 6

   
   

9. r(x) = x + 5 and q(x) = 2x² − 5, Find q(r(−2))
   a. 8 c. 13
   b. b. -8 d. -13

   
   

10. Let f(x) = 3x + 8 and g(x) = x − 2. Find f(g(x)).
    a. 2x + 3 c. 4x + 1
    b. 2x − 3 d. 3x + 2

    
    

For numbers 11-13, refer to the figure below:

[chart: mapping diagram; figure for numbers 11-13, with r(x) = 2x − 7 and s(x) = (2x − 5)/(x + 2)]

Domain	r(x) = 2x − 7	s(x) = (2x − 5)/(x + 2)
−2	−11	3
1	−5	5
2	−3	11
3	−1	−7
4	1	−1
7	7	1

[transcriber’s note: the values are read from the horizontal arrows connecting each row of the figure]

11. Evaluate r(2)
    a. -11 c. 5
    b. -3 d. 11

    
    

12. Find s(r(7))
    a. 7 c. -1
    b. 1 d. -7

    
    

13. Find (s ∘ r)(1)
    a. -3 c. 5
    b. 3 d. -5

    
    

For numbers 14-15, refer to the table of values below

t(x) = 2x + 1 k(x) = 2x² − 7x − 5

x	0	1	2	3	4	5	6	7	8
t(x)	1	3	5	7	9	11	13	15	17
k(x)	-5	-10	-11	-8	-1	10	25	44	67

14. Find (k − t)(4)
    a. 8 c. 10
    b. -8 d. -10

    
    

15. Find (k ∘ t)(2)
    a. 10 c. -5
    b. b. -10 d. -1

    
    

Additional Activities

PUNCH D LINE

Find out some of favorite punch lines by answering operations on functions problems below. Phrases of punch lines were coded by the letters of the correct answers. Write the punch lines on the lines provided.

Given: f(x) = 2x − 1 g(x) = |3x − 4| h(x) = x/2 r(x) = x + 3 s(x) = x² − 4x − 21

Column A	Column B
_______1. f(0) =	A. −11
_______2. g(3) =	B. 2
_______3. s(−1) =	C. 3x + 2
_______4. h(0) =	D. x − 7
_______5. (f + r)(x) =	E. −x + 4
_______6. (f + r)(3) =	F. 0
_______7. (r − f)(x) =	G. 2x² + 5x − 3
_______8. (r − f)(2) =	H. 6
_______9. (f • r)(x) =	I. −16
_______10. (f • r)(1) =	J. 2x + 2
_______11. (s/r)(x) =	K. 5
_______12. (s/r)(−4) =	L. 1
_______13. (r ∘ f)(x) =	M. 11
_______14. (r ∘ f)(2) =	N. −1
_______15. (g ∘ f)(1) =	O. 4

Code:

Tingnan mo ako	K	ang laman ng utak ko?	J
para may attachment lagi tayo	L	Buhay nga pero patay	I
ang parents ko	E	Hindi lahat ng buhay ay buhay	N
na ako sa’yo	O	Di mo pa nga ako binabato	B
Masasabi mo bang bobo ako?	D	na patay naman sa’yo	F
Kasi, botong-boto sayo	M	Tatakbo ka ba sa eleksyon?	C
Kung ikaw lamang	A	pero tinamaan	G
Sana naging email na lang ako	H

Punch lines: (1-4) ___________________________________________________________ (5-7) ___________________________________________________________ (8-10) ___________________________________________________________ (11-13) ___________________________________________________________ (14-15) ___________________________________________________________

Answer Key

[transcriber’s note: this page is printed upside down in the source; the text below is given in upright reading order]

What’s More

Activity 1: Matching Functions

1. g 6. c
2. j 7. i
3. e 8. b
4. a 9. h
5. f 10. d

Activity 2: Let’s Simplify

A.

1. (m − p)(x) = −2x² − 3x + 2
2. p(5) + m(3) − h(1) = 79
3. m(x)/h(x) = 1/(x+3) [?]
4. p(x + 1) = 2x² + 9x + 4
5. p(3) − 3m(2) = 21

B.

1. (m + n)(x) = 6x + 1
2. (m · n)(x) = 5x² + 17x − 12
3. (n − c)(x) = −5x² − 16x + 16
4. (c/m)(x) = x + 4 [?]
5. (m ∘ n)(x) = 5x + 17
6. (n ∘ c)(−3) = −14
7. n(m(m(2))) = 38

C.

1. p(x) = g(x)/h(x)
2. r(x) = g(x) + h(x)
3. s(x) = g(x) • h(x)
4. t(x) = h(x) − g(x)

D.

1. 3x² − 16x + 17
2. a. 2x² + 7x − 15
   b. 1/(x−2)
   c. x² + 13x + 30
3. 2
4. 12
5. (6x² + 26x − 16)/(x + 5)
6. 6x³ − 27x² − 15x

What I Know

1. b	6. c	11. c
2. d	7. c	12. d
3. a	8. a	13. c
4. b	9. d	14. a
5. a	10. d	15. d

What’s In

1. N	6. C	11. L	16. G
2. E	7. A	12. U	17. M
3. I	8. H	13. O
4. W	9. V	14. S
5. T	10. R	15. D

Secret Message: WELCOME TO SENIOR HIGH SCHOOL IM GLAD YOU ARE HERE

What’s New

a. After 30 days: Neah has ₱160 and Neoh has ₱95 After 365 days or 1 year: Neah has ₱1835 and Neoh has ₱1100 Their combined savings for 1 year is ₱2935

b. Answers may vary c. Answers may vary d. Answers may vary e. Answers may vary f. Let x = number of days f(x) = amount of savings of Neah g(x) = amount of savings of Neoh f(x) = 5x + 10 g(x) = 3x + 5

What I Can Do

a. 1. R(I(3)) = R(12) = 5 On the third day, there were 12 infected and 5 recovered people

2. R(I(8)) = R(82) = 25
   On 8th day, there were 82 people infected and 25 recovered people.

   
   

3. I(R(18)) = I(7) = 47
   Although we can evaluate the composition of function here, this value does not make sense. I(d) is the function of days, but 7 in I(7) means number of recovered people.

   
   

b. Answers may vary

1. M(I(1)) = M(5) = 0
   On the first day, there were 5 infected and no death.

   
   

2. M(I(4)) = M(18) = 1
   On the fourth day, there were 18 infected and 1 death

   
   

3. I(M(12)) = I(1) = 5
   Although we can evaluate the composition of function here, this value does not make sense. I(d) is the function of days, but 1 in I(1) means number of deaths.

   
   

c. Answers may vary

d. Answers may vary

Assessment

1. b	6. d	11. b
2. d	7. a	12. b
3. c	8. b	13. c
4. c	9. c	14. d
5. d	10. d	15. A

Additional Activities

Punch d line

1. N	6. M	11. D
2. K	7. E	12. A
3. I	8. B	13. J [?]
4. F	9. G	14. H
5. C	10. O	15. L

Punch lines:

1. Hindi lahat ng buhay ay buhay, tingnan mo ako, buhay nga pero patay na patay naman sa’yo.

   
   

2. Taalkbo ke ba sa elekeyon? Kasi, botong-boto sayo ang parents ko.

   
   

3. Di mo pa alam ako pero tinamaan na ako sa’yo.

   
   

4. Masabi mo bang bobo ako? Kung ikaw lamang ang laman ng utak ko?

   
   

5. Sana naging email na lang ako para may attachment lagi tayo.

   
   

References

Department of Education. “General Mathematics Learner’s Material.” In General Mathematics Learner’s Material, by Debbie Marie B. Verzosa, Paolo L. Apolinario, Regina M. Tresvalles, Francis Nelson M. Infante, Jose Lorenzo M. Sin and Len Patrick Dominic M. Garces, edited by Leo Andrei A. Crisologo, Shirlee R. Ocampo, Jude Buot, Lester C. Hao, Eden Delight P. Miro and Eleanor Villanueva, 13-20. Meralco Avenue, Pasig City, Philippines 1600: Lexicon Pres Inc., 2016.

Department of education. “General Mathematics Teacher’s Guide.” In General Mathematics Teacher’s Guide, by Leo Andrei A. Crisologo, Shirlee R. Ocampo, Eden Delight P. Miro, Regina M. Tresvalles, Lester C. Hao and Emellie G. Palomo, edited by Christian Paul O. Chan Shio and Mark L. Loyola, 14-22. Meralco Avenue, Pasig City, Philippines 1600: Lexicon Press Inc., 2016.

coronatracker.com. COVID-19 Corona Tracker. n.d. https://www.coronatracker.com/country/philippines/ (accessed May 20, 2020).

engageny.org. n.d. https://www.engageny.org/file/128826/download/precalculus-m3-topic-b-lesson-16-teacher.pdf?token=pvy6pn0x (accessed May 20, 2020).

quizizz.com. n.d. https://quizizz.com/admin/search/operations%20on%20functions (accessed May 22, 2020).

For inquiries or feedback, please write or call:

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