Downloaded 35 times
![Composite Function:
f g x f [gx]
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.](https://image.slidesharecdn.com/operationoffunctionsandcompositefunction-220921005902-5fbe4cf8/85/Operation-of-functions-and-Composite-function-pptx-25-320.jpg)
![Composite Function:
Example 1: 𝒇 𝒙 = 𝟒𝒙 + 𝟏𝟎 𝒂𝒏𝒅 𝒈 𝒙 = 𝒙 + 𝟏
= 𝟒(𝒙 + 𝟏) + 𝟏𝟎
𝟒𝒙 +𝟒 +𝟏𝟎
𝟒𝒙 + 𝟏𝟒
f [gx]=𝟒𝒙 + 𝟏𝟒](https://image.slidesharecdn.com/operationoffunctionsandcompositefunction-220921005902-5fbe4cf8/85/Operation-of-functions-and-Composite-function-pptx-26-320.jpg)
![Another one…
Given that: 𝒇 𝒙 = 𝟖𝒙 + 𝟐 𝒂𝒏𝒅 𝒈 𝒙 = −𝟑𝒙 − 𝟕, 𝒇𝒊𝒏𝒅:
𝟏. 𝒇𝒈 𝒙 2. 𝒇𝒈 −𝟐
𝟖(−𝟑𝒙 − 𝟕) + 𝟐
−𝟐𝟒𝒙 − 𝟓𝟔 + 𝟐
f [gx]=−𝟐𝟒𝒙−𝟓𝟒
−𝟐𝟒𝒙 − 𝟓𝟒
−𝟐𝟒(−𝟐) − 𝟓𝟒
𝟒𝟖 − 𝟓𝟒
𝒇[𝒈 −𝟐 ]=−𝟔](https://image.slidesharecdn.com/operationoffunctionsandcompositefunction-220921005902-5fbe4cf8/85/Operation-of-functions-and-Composite-function-pptx-28-320.jpg)
More Related Content
Similar to Operation of functions and Composite function.pptx
Operation of functions and Composite function.pptx
- 1. Prepared by: Mr. GeorgeG. Lescano
- 3. Addition of Function: Tip1: Combining like terms. Tip 3: Be careful with the integers. Tip 2: Put the equation/s in DESCENDING ORDERS.
- 4. Addition of Function: 𝒇𝒙 = 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: Example2: 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: Usingthe 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 gx f x gx CAUTION: Make sure you distribute the – to each term of the second function. You should simplify by combining like terms.
- 8. Subtraction of Function: Example1: 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: Example2: 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: 𝒈𝒙 = −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* gx f x* gx 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: 𝒇𝒙 = −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: Example2: 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: 𝒔𝒙 = 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: Whenyou 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: Polynomiallong- division Synthetic division
- 19. Example 1: 𝒇𝒙 = 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: Usingthe 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 functionor 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 [gx] 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 [gx]=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. 𝒇𝒈 𝒙 2. 𝒇𝒈 −𝟐 𝟖(−𝟑𝒙 − 𝟕) + 𝟐 −𝟐𝟒𝒙 − 𝟓𝟔 + 𝟐 f [gx]=−𝟐𝟒𝒙−𝟓𝟒 −𝟐𝟒𝒙 − 𝟓𝟒 −𝟐𝟒(−𝟐) − 𝟓𝟒 𝟒𝟖 − 𝟓𝟒 𝒇[𝒈 −𝟐 ]=−𝟔
- 29. Quotation of theday: “ One of the lesson of math in our life that we should to apply is always be careful with the sign ” - Anonymous