General Mathematics Operation of Function and Composite Function

1. Prepared by:
Mr. George G. Lescano

3. Addition of Function:
Tip 1: Combining like terms.
Tip 3: Be careful with the integers.
Tip 2: Put the equation/s in DESCENDING
ORDERS.

4. Addition of Function:
𝒇 𝒙 = 𝒙𝟐 − 𝟑 𝒂𝒏𝒅 𝒈 𝒙 = 𝟐𝒙 + 𝟓
Example 1: −𝟑 +5
𝒙𝟐 −3
𝟐𝒙 +𝟓
+
+𝟐
𝒙𝟐+2x
𝒙𝟐 + 𝟐𝒙 − 𝟑 + 𝟓
𝒙𝟐
+ 𝟐𝒙 + 𝟐
𝑓 + 𝑔 𝑥 = 𝑥2
+ 2𝑥 + 2

5. Addition of Function:
Example 2:
𝒂 𝒙 = 𝒙𝟐 − 𝟑𝒙 + 𝟏 − 𝟐𝒙𝟑 𝒂𝒏𝒅 𝒃 𝒙 = −𝟓𝒙 + 𝒙𝟑 − 𝟏𝟎
𝒂 𝒙 = −𝟐𝒙𝟑
+ 𝒙𝟐
− 𝟑𝒙 + 𝟏 𝒃 𝒙 = 𝒙𝟑
− 𝟓𝒙 − 𝟏𝟎
−𝟐𝒙𝟑 + 𝒙𝟐 − 𝟑𝒙 + 𝟏
𝒙𝟑
+ −𝟓𝒙 − 𝟏𝟎
−𝒙𝟑 + 𝒙𝟐 −𝟖𝒙 −𝟗
𝒂 + 𝒃 𝒙 = −𝒙𝟑
+ 𝒙𝟐
− 𝟖𝒙 − 𝟗

6. Addition of Function:
Using the same given in example 2, find:
𝒂 𝒙 = 𝒙𝟐
− 𝟑𝒙 + 𝟏 − 𝟐𝒙𝟑
𝒂𝒏𝒅 𝒃 𝒙 = −𝟓𝒙 + 𝒙𝟑
− 𝟏𝟎
(𝒂 + 𝒃)(−𝟑)
𝒂 + 𝒃 −𝟑 = 𝟓𝟏
𝒂 + 𝒃 𝒙 = −𝒙𝟑
+ 𝒙𝟐
− 𝟖𝒙 − 𝟗
= − −𝟑 𝟑
+ −𝟑 𝟐
− 𝟖(−𝟑) − 𝟗
= − −𝟐𝟕 + 𝟗 + 𝟐𝟒 − 𝟗
= 𝟐𝟕 + 𝟗 + 𝟐𝟒 − 𝟗

7. Subtraction of Function:
 f gx f 
x gx
CAUTION: Make sure you distribute the – to each
term of the second function. You should simplify
by combining like terms.

8. Subtraction of Function:
Example 1: 𝒇 𝒙 = 𝟑𝒙𝟐
+ 𝟏𝟎 𝒂𝒏𝒅 𝒈 𝒙 = 𝟒𝒙 + 𝟕
𝟑𝒙𝟐 + 𝟏𝟎 (𝟒𝒙 + 𝟕)
−
𝟑𝒙𝟐
+ 𝟏𝟎 −𝟒𝒙 −𝟕
𝟑𝒙𝟐
− 𝟒𝒙 + 𝟏𝟎 − 𝟕
𝟑𝒙𝟐
− 𝟒𝒙 + 𝟑
𝒇 − 𝒈 𝒙 = 𝟑𝒙𝟐 − 𝟒𝒙 + 𝟑

9. Subtraction of Function:
Example 2: 𝒈 𝒙 = −𝒙𝟐
+ 𝟗 −𝟐𝒙𝟑
𝒂𝒏𝒅 𝒉 𝒙 = −𝟑𝒙 + 𝟒
−𝟐𝒙𝟑 −𝒙𝟐 +𝟗 (−𝟑𝒙 + 𝟒)
−
+𝟑𝒙 −𝟒
−𝟐𝒙𝟑 −𝒙𝟐 +𝟑𝒙 + 𝟗 − 𝟒
𝒈 − 𝒉 𝒙 = −𝟐𝒙𝟑
−𝒙𝟐
+𝟑𝒙 + 𝟓
−𝟐𝒙𝟑 −𝒙𝟐 +𝟗
−𝟐𝒙𝟑
−𝒙𝟐
+𝟑𝒙 + 𝟓

10. Subtraction of Function:
𝒈 𝒙 = −𝒙𝟐 + 𝟗 −𝟐𝒙𝟑 𝒂𝒏𝒅 𝒉 𝒙 = −𝟑𝒙 + 𝟒
Using the same given in example 2, find:
(𝒈 − 𝒉)(𝟐)
𝒈 − 𝒉 𝒙 = −𝟐𝒙𝟑 −𝒙𝟐 +𝟑𝒙 + 𝟓
= −𝟐(𝟐)𝟑 −(𝟐)𝟐 +𝟑 𝟐 + 𝟓
= −𝟐 𝟖 − (𝟒) + 𝟔 + 𝟓
= −𝟏𝟔 − 𝟒 + 𝟏𝟏 𝒈 − 𝒉 𝟐 = −𝟗

11. Multiplication of Function:
f * gx f 
x* gx
To find the product of two functions, put
parenthesis around them and multiply
each term from the first function to each
term of the second function.

12. Multiplication of Function:
𝒇 𝒙 = −𝟕𝒙 + 𝟏𝟐 𝒂𝒏𝒅 𝒈 𝒙 = 𝟐𝒙 − 𝟓
Example 1:
−𝟕𝒙 + 𝟏𝟐
𝟐𝒙 − 𝟓
∗
−𝟏𝟒𝒙𝟐
+ 𝟐𝟒𝒙
𝟑𝟓𝒙 − 𝟔𝟎
−𝟏𝟒𝒙𝟐 +𝟓𝟗𝒙 − 𝟔𝟎
(−𝟕𝒙 + 𝟏𝟐)(𝟐𝒙 − 𝟓)
𝑭 =
𝑶 =
𝑰 =
𝑳 =
−𝟏𝟒𝒙𝟐
𝟑𝟓𝒙
𝟐𝟒𝒙
−𝟔𝟎
𝟓𝟗𝒙
𝒇 ∗ 𝒈 𝒙 = −𝟏𝟒𝒙𝟐
+ 𝟓𝟗𝒙 − 𝟔𝟎

13. Multiplication of Function:
Example 2: 𝒔 𝒙 = 𝟐𝒙𝟐 − 𝟐 +𝟑𝒙𝟑 𝒂𝒏𝒅 𝒕 𝒙 = −𝒙 + 𝟏
𝟑𝒙𝟑 +𝟐𝒙𝟐 −𝟐
−𝒙 + 𝟏
−𝟑𝒙𝟒
−𝟐𝒙𝟑
+𝟐𝒙
𝟑𝒙𝟑
+𝟐𝒙𝟐
−𝟐
−𝟑𝒙𝟒
+𝒙𝟑
+𝟐𝒙𝟐
+ 𝟐𝒙 − 𝟐
𝒔 ∗ 𝒕 𝒙 = −𝟑𝒙𝟒 +𝒙𝟑 +𝟐𝒙𝟐 + 𝟐𝒙 − 𝟐

14. Multiplication of Function:
𝒔 𝒙 = 𝟐𝒙𝟐
− 𝟐 +𝟑𝒙𝟑
𝒂𝒏𝒅 𝒕 𝒙 = −𝒙 + 𝟏
Using the same given in example 2, find:
(𝒔 ∗ 𝒕)
𝟏
𝟐
𝒔 ∗ 𝒕 𝒙 = −𝟑𝒙𝟒 +𝒙𝟑 +𝟐𝒙𝟐 + 𝟐𝒙 − 𝟐
= −𝟑
𝟏
𝟐
𝟒
+
𝟏
𝟐
𝟑
+𝟐
𝟏
𝟐
𝟐
+ 𝟐
𝟏
𝟐
− 𝟐

15. Multiplication of Function:
= −𝟑
𝟏
𝟐
𝟒
+
𝟏
𝟐
𝟑
+𝟐
𝟏
𝟐
𝟐
+ 𝟐
𝟏
𝟐
− 𝟐
= −𝟑
𝟏
𝟏𝟔
+
𝟏
𝟖
+ 𝟐
𝟏
𝟒
+ 𝟐
𝟏
𝟐
− 𝟐
=
−𝟑
𝟏𝟔
+
𝟏
𝟖
+
𝟏
𝟐
+ 𝟏 − 𝟐
𝒔 ∗ 𝒕
𝟏
𝟐
=
−𝟗
𝟏𝟔

16. Multiplication of Function:
𝒔 ∗ 𝒕
𝟏
𝟐
=
−𝟗
𝟏𝟔

17. Division of Function:
When you divide two such functions together,
you get what is called a rational expression.
A rational expression is the division of two
polynomials. If they divide evenly, your answer
will become a polynomial.

18. Division of Function:
Polynomial long- division
Synthetic division

19. Example 1: 𝒇 𝒙 = 𝟑𝒙𝟐
+ 𝟒𝒙 + 𝟓 𝒂𝒏𝒅 𝒈 𝒙 = 𝒙 + 𝟐
𝟑𝒙𝟐 + 𝟒𝒙 + 𝟓
𝒙 + 𝟐
𝟑𝒙
𝟑𝒙𝟐
− 𝟔𝒙
−
−𝟐𝒙 +𝟓
−𝟐
−
𝟐𝒙 +𝟒
𝟗
𝒇
𝒈
𝒙 = 𝟑𝒙 − 𝟐 +
𝟗
𝒙 + 𝟐

20. Example: 𝒇 𝒙 = 𝟑𝒙𝟐
+ 𝟒𝒙 + 𝟓 𝒂𝒏𝒅 𝒈 𝒙 = 𝒙 + 𝟐
𝑼𝒔𝒊𝒏𝒈 𝑺𝒚𝒏𝒕𝒉𝒆𝒕𝒊𝒄 𝒅𝒊𝒗𝒊𝒔𝒊𝒐𝒏:
𝒙 + 𝟐 = 𝟎
𝒙 + 𝟐 − 𝟐 = 𝟎 − 𝟐
−𝟐
𝒙 = −𝟐
𝟑 𝟒 𝟓
𝟑
−𝟔
−𝟐
𝟒
𝟗
𝒇
𝒈
𝒙 = 𝟑𝒙 − 𝟐 +
𝟗
𝒙 + 𝟐
𝒓𝒆𝒎𝒂𝒊𝒏𝒅𝒆𝒓

21. Division of Function:
Using the same given in the example, find:
𝒇 𝒙 = 𝟑𝒙𝟐 + 𝟒𝒙 + 𝟓 𝒂𝒏𝒅 𝒈 𝒙 = 𝒙 + 𝟐
𝒇
𝒈
𝟏
𝟐
𝒇
𝒈
𝒙 = 𝟑𝒙 − 𝟐 +
𝟗
𝒙 + 𝟐
= 𝟑
𝟏
𝟐
− 𝟐 + 𝟗
=
𝟑
𝟐
+ 𝟕
𝒇
𝒈
𝟏
𝟐
=
𝟏𝟕
𝟐
𝒐𝒓𝟖
𝟏
𝟐

24. Composite Function:
Composite function or composition of
function is another way of combining
function.
This method of combining function uses the
output of one function as the input for a
second function.

25. Composite Function:
f  g x f [gx]
This is read “f composition g” or “f composed
g” and means to copy the f function down but
where ever you see an x, substitute in the g
function.

26. Composite Function:
Example 1: 𝒇 𝒙 = 𝟒𝒙 + 𝟏𝟎 𝒂𝒏𝒅 𝒈 𝒙 = 𝒙 + 𝟏
= 𝟒(𝒙 + 𝟏) + 𝟏𝟎
𝟒𝒙 +𝟒 +𝟏𝟎
𝟒𝒙 + 𝟏𝟒
f [gx]=𝟒𝒙 + 𝟏𝟒

27. Composite Function:
Example 2: 𝒉 𝒙 = 𝟑𝒙𝟐 − 𝒙 + 𝟖 𝒂𝒏𝒅 𝒌 𝒙 = −𝟐𝒙 + 𝟑
𝟑(−𝟐𝒙 + 𝟑 )𝟐 − (−𝟐𝒙 + 𝟑) + 𝟖
𝟑(𝟒𝒙𝟐
− 𝟏𝟐𝒙 + 𝟗)+𝟐𝒙 − 𝟑 +𝟖
𝟏𝟐𝒙𝟐
− 𝟑𝟔𝒙 + 𝟐𝟕 + 𝟐𝒙 − 𝟑 + 𝟖
ℎ 𝑘 𝑥 = 12𝑥2
− 34𝑥 + 32

28. Another one…
Given that: 𝒇 𝒙 = 𝟖𝒙 + 𝟐 𝒂𝒏𝒅 𝒈 𝒙 = −𝟑𝒙 − 𝟕, 𝒇𝒊𝒏𝒅:
𝟏. 𝒇𝒈 𝒙 2. 𝒇𝒈 −𝟐
𝟖(−𝟑𝒙 − 𝟕) + 𝟐
−𝟐𝟒𝒙 − 𝟓𝟔 + 𝟐
f [gx]=−𝟐𝟒𝒙−𝟓𝟒
−𝟐𝟒𝒙 − 𝟓𝟒
−𝟐𝟒(−𝟐) − 𝟓𝟒
𝟒𝟖 − 𝟓𝟒
𝒇[𝒈 −𝟐 ]=−𝟔

29. Quotation of the day:
“ One of the lesson of math in our life
that we should to apply is always be
careful with the sign ”
- Anonymous