Operations on functions can include addition, subtraction, multiplication, division, and composition. Adding two functions results in a function where the values are added at each point. Multiplying functions results in a function where the values are multiplied at each point. Composing functions means applying one function to another, resulting in another function.

1. Operation on Functions

2. 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.

3. ACTIVITY: SECRET MESSAGE
Direction. 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 and . A. (x + 4)(x − 3)
3 1 4x+7
_____2. Find the LCD of x−2 and x+3 C. x 2+x−6
_____3. Find the sum of and . D.
2 5 2 + x − 6
_____4. Find the sum of + E. (𝑥 − 2)(𝑥 + 3) or x x x
𝑥+4

4. 5. Find the product of and. G.
x+2
3 1
_____6. Find the sum of and H. (x + 1)(x − 6) x−2 x+3
For numbers 7-14, find the factors.
_____7. x2 + x − 12 I.
_____8. x2 − 5x − 6 L. (𝑥 − 4(𝑥 − 3)
_____9. x2 + 6x + 5 M. −5
_____10. x2 + 7x + 12 N. 21
_____11. x2 − 7x + 12 O. (𝑥 − 5)(𝑥 − 3)
_____12. x2 − 5x − 14 R. (x + 4)(x + 3)
_____13. x2 − 8x + 15 S. (𝑥 − 7)(𝑥 − 5)
_____14. x2 − 12x + 35 T.
x2+x−12 x2+6x+5
_____15. Find the product of x2−5x−6 and x2+7x+12. U. (𝑥 − 7(𝑥 + 2)
x2−5x−14 x2−12x+35 𝑥
_____17. In the function f(x) = 4 − x2, 𝑓𝑖𝑛𝑑 𝑓(−3) Y. (x + 5)(x + 1)
x2+x−12 x2−8x+15 7 _____16. Divide by W.

5. Secret Message:
16 11 7 15 9 13 12 7 10 2 8 2 10 2

6. Definition. Let f and g be functions.
• Their sum, denoted by 𝑓 + 𝑔, is the function denoted by (𝑓 + 𝑔)(𝑥) =
𝑓(𝑥) + 𝑔(𝑥).
• Their difference, denoted by 𝑓 − 𝑔, is the function denoted by (𝑓 − 𝑔
)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥).
• Their product, denoted by 𝑓 • 𝑔, is the function denoted by (𝑓 • 𝑔)(𝑥)
= 𝑓(𝑥) • 𝑔(𝑥).
• 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.
• The composite function denoted by (𝑓 ° 𝑔)(𝑥) = 𝑓(𝑔(𝑥)). The process
of obtaining a composite function is called function composition.

7. Example 1. Given the functions:
•𝑓(𝑥) = 𝑥 + 5 𝑔(𝑥) = 2𝑥 − 1 ℎ(𝑥) = 2𝑥2 + 9𝑥 − 5
•Determine the following functions:
a. (𝑓 + 𝑔)(𝑥) 𝑒. (𝑓 + 𝑔)(3)
b. (𝑓 − 𝑔)(𝑥) 𝑓. (𝑓 − 𝑔)(3)
c. (𝑓 • 𝑔)(𝑥) 𝑔. (𝑓 • 𝑔)(3)
d. (
ℎ
𝑔
)(𝑥) h. (
ℎ
𝑔
)3 =
ℎ(3)
𝑔(3)

8. 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

9. b. (𝑓 • 𝑔)(𝑥) = 𝑓(𝑥) • 𝑔(𝑥) definition of multiplication of functions
= (𝑥 + 5) • (2𝑥 − 1) replace f(x) and g(x) by the given values
= 2𝑥2 + 9𝑥 − 5 multiply the binomials
c. (ℎ)(𝑥) = definition of division of functions
𝑔
= replace h(x) and g(x) by the given values
= factor the numerator
= cancel out common factors
= 𝑥 + 5

10. b. (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
∴ 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

11. 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 • 𝑔)(3) = 𝑓(3) • 𝑔(3) definition of multiplication of functions
= 8 • 5 replace f(3) and g(3) by the given values
= 40
Alternative solution:
multiply
(𝑓 • 𝑔)(𝑥) = 2𝑥2 + 9𝑥 − 5 resulted function from item c
(𝑓 • 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

12. h. (ℎ)(3) = ℎ (3)
𝑔 𝑔(3)
Solve for ℎ(3) and 𝑔(3) separately:
ℎ(𝑥) = 2𝑥2 + 9𝑥 − 5 𝑔(𝑥) = 2𝑥 − 1 ℎ(3)
= 2(3)2 + 9(3) − 5 𝑔(3) = 2(3) − 1
= 18 + 27 − 5 = 5
= 40
Alternative solution:
(ℎ)(𝑥) = 𝑥 + 5 resulted function from item d
𝑔
(h)(x) = 3 + 5 replace x by 3 g
= 8 simplify

13. ILLUSTRATIONS
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.

14. ADDITION

15. SUBTRACTION

16. MULTIPLICATION

17. DIVISION

18. 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”.

19. Example 2. Given 𝑓(𝑥) = 𝑥2 + 5𝑥 + 6, and ℎ(𝑥) = 𝑥 + 2
Find the following:
a. (𝑓 ∘ ℎ)(𝑥)
b. (𝑓 ∘ ℎ)(4)
c. (ℎ ∘ 𝑓)(𝑥)
Solution.
a. (𝑓 ∘ ℎ)(𝑥) = 𝑓(ℎ(𝑥)) definition of function composition
= 𝑓(𝑥 + 2) replace h(x) by x+2
Since 𝑓(𝑥) = 𝑥2 + 5𝑥 + 6 given
𝑓(𝑥 + 2) = ()2 + 5(𝑥 + 2) + 6 replace x by x+2
= 𝑥2 + 4𝑥 + 4 + 5𝑥 + 10 + 6 perform the operations
= 𝑥2 + 9𝑥 + 20 combine similar terms
𝑥 + 2

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

21. b. (𝑓 ∘ ℎ)(4) = 𝑓(ℎ(4))
Step 1. Evaluate ℎ(4) Step 2. Evaluate 𝑓(6) ℎ(𝑥) = 𝑥 + 2 𝑓(𝑥)
= 𝑥2 + 5𝑥 + 6 ℎ(4) = 4 + 2 𝑓(6) = 62 + 5(6) + 6
= 6 = 36 + 30 + 6
= 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:
definition of function composition
𝑓 , from item a
replace all x’s by 4
perform the indicated operations
simplify

23. ACTIVITY: PERFORM THE INDICATED OPERATIONS
a= 3
n= 3
t= 3