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Operation of Function and Composite Function

Education
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Prepared by: Mr. George G. Lescano
Addition of Function: Tip 1: Combining like terms. Tip 3: Be careful with the integers. Tip 2: Put the equation/s in DESCENDING ORDERS.
Addition of Function: 𝒇 𝒙 = π’™πŸ βˆ’ πŸ‘ 𝒂𝒏𝒅 π’ˆ 𝒙 = πŸπ’™ + πŸ“ Example 1: βˆ’πŸ‘ +5 π’™πŸ βˆ’3 πŸπ’™ +πŸ“ + +𝟐 π’™πŸ +2x π’™πŸ + πŸπ’™ βˆ’ πŸ‘ + πŸ“ π’™πŸ + πŸπ’™ + 𝟐 𝑓 + 𝑔 π‘₯ = π‘₯2 + 2π‘₯ + 2
Addition of Function: Example 2: 𝒂 𝒙 = π’™πŸ βˆ’ πŸ‘π’™ + 𝟏 βˆ’ πŸπ’™πŸ‘ 𝒂𝒏𝒅 𝒃 𝒙 = βˆ’πŸ“π’™ + π’™πŸ‘ βˆ’ 𝟏𝟎 𝒂 𝒙 = βˆ’πŸπ’™πŸ‘ + π’™πŸ βˆ’ πŸ‘π’™ + 𝟏 𝒃 𝒙 = π’™πŸ‘ βˆ’ πŸ“π’™ βˆ’ 𝟏𝟎 βˆ’πŸπ’™πŸ‘ + π’™πŸ βˆ’ πŸ‘π’™ + 𝟏 π’™πŸ‘ + βˆ’πŸ“π’™ βˆ’ 𝟏𝟎 βˆ’π’™πŸ‘ + π’™πŸ βˆ’πŸ–π’™ βˆ’πŸ— 𝒂 + 𝒃 𝒙 = βˆ’π’™πŸ‘ + π’™πŸ βˆ’ πŸ–π’™ βˆ’ πŸ—
Addition of Function: Using the same given in example 2, find: 𝒂 𝒙 = π’™πŸ βˆ’ πŸ‘π’™ + 𝟏 βˆ’ πŸπ’™πŸ‘ 𝒂𝒏𝒅 𝒃 𝒙 = βˆ’πŸ“π’™ + π’™πŸ‘ βˆ’ 𝟏𝟎 (𝒂 + 𝒃)(βˆ’πŸ‘) 𝒂 + 𝒃 βˆ’πŸ‘ = πŸ“πŸ 𝒂 + 𝒃 𝒙 = βˆ’π’™πŸ‘ + π’™πŸ βˆ’ πŸ–π’™ βˆ’ πŸ— = βˆ’ βˆ’πŸ‘ πŸ‘ + βˆ’πŸ‘ 𝟐 βˆ’ πŸ–(βˆ’πŸ‘) βˆ’ πŸ— = βˆ’ βˆ’πŸπŸ• + πŸ— + πŸπŸ’ βˆ’ πŸ— = πŸπŸ• + πŸ— + πŸπŸ’ βˆ’ πŸ—
Subtraction of Function:  fο€­ 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.
Subtraction of Function: Example 1: 𝒇 𝒙 = πŸ‘π’™πŸ + 𝟏𝟎 𝒂𝒏𝒅 π’ˆ 𝒙 = πŸ’π’™ + πŸ• πŸ‘π’™πŸ + 𝟏𝟎 (πŸ’π’™ + πŸ•) βˆ’ πŸ‘π’™πŸ + 𝟏𝟎 βˆ’πŸ’π’™ βˆ’πŸ• πŸ‘π’™πŸ βˆ’ πŸ’π’™ + 𝟏𝟎 βˆ’ πŸ• πŸ‘π’™πŸ βˆ’ πŸ’π’™ + πŸ‘ 𝒇 βˆ’ π’ˆ 𝒙 = πŸ‘π’™πŸ βˆ’ πŸ’π’™ + πŸ‘
Subtraction of Function: Example 2: π’ˆ 𝒙 = βˆ’π’™πŸ + πŸ— βˆ’πŸπ’™πŸ‘ 𝒂𝒏𝒅 𝒉 𝒙 = βˆ’πŸ‘π’™ + πŸ’ βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ— (βˆ’πŸ‘π’™ + πŸ’) βˆ’ +πŸ‘π’™ βˆ’πŸ’ βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ‘π’™ + πŸ— βˆ’ πŸ’ π’ˆ βˆ’ 𝒉 𝒙 = βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ‘π’™ + πŸ“ βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ— βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ‘π’™ + πŸ“
Subtraction of Function: π’ˆ 𝒙 = βˆ’π’™πŸ + πŸ— βˆ’πŸπ’™πŸ‘ 𝒂𝒏𝒅 𝒉 𝒙 = βˆ’πŸ‘π’™ + πŸ’ Using the same given in example 2, find: (π’ˆ βˆ’ 𝒉)(𝟐) π’ˆ βˆ’ 𝒉 𝒙 = βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ‘π’™ + πŸ“ = βˆ’πŸ(𝟐)πŸ‘ βˆ’(𝟐)𝟐 +πŸ‘ 𝟐 + πŸ“ = βˆ’πŸ πŸ– βˆ’ (πŸ’) + πŸ” + πŸ“ = βˆ’πŸπŸ” βˆ’ πŸ’ + 𝟏𝟏 π’ˆ βˆ’ 𝒉 𝟐 = βˆ’πŸ—
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.
Multiplication of Function: 𝒇 𝒙 = βˆ’πŸ•π’™ + 𝟏𝟐 𝒂𝒏𝒅 π’ˆ 𝒙 = πŸπ’™ βˆ’ πŸ“ Example 1: βˆ’πŸ•π’™ + 𝟏𝟐 πŸπ’™ βˆ’ πŸ“ βˆ— βˆ’πŸπŸ’π’™πŸ + πŸπŸ’π’™ πŸ‘πŸ“π’™ βˆ’ πŸ”πŸŽ βˆ’πŸπŸ’π’™πŸ +πŸ“πŸ—π’™ βˆ’ πŸ”πŸŽ (βˆ’πŸ•π’™ + 𝟏𝟐)(πŸπ’™ βˆ’ πŸ“) 𝑭 = 𝑢 = 𝑰 = 𝑳 = βˆ’πŸπŸ’π’™πŸ πŸ‘πŸ“π’™ πŸπŸ’π’™ βˆ’πŸ”πŸŽ πŸ“πŸ—π’™ 𝒇 βˆ— π’ˆ 𝒙 = βˆ’πŸπŸ’π’™πŸ + πŸ“πŸ—π’™ βˆ’ πŸ”πŸŽ
Multiplication of Function: Example 2: 𝒔 𝒙 = πŸπ’™πŸ βˆ’ 𝟐 +πŸ‘π’™πŸ‘ 𝒂𝒏𝒅 𝒕 𝒙 = βˆ’π’™ + 𝟏 πŸ‘π’™πŸ‘ +πŸπ’™πŸ βˆ’πŸ βˆ’π’™ + 𝟏 βˆ’πŸ‘π’™πŸ’ βˆ’πŸπ’™πŸ‘ +πŸπ’™ πŸ‘π’™πŸ‘ +πŸπ’™πŸ βˆ’πŸ βˆ’πŸ‘π’™πŸ’ +π’™πŸ‘ +πŸπ’™πŸ + πŸπ’™ βˆ’ 𝟐 𝒔 βˆ— 𝒕 𝒙 = βˆ’πŸ‘π’™πŸ’ +π’™πŸ‘ +πŸπ’™πŸ + πŸπ’™ βˆ’ 𝟐
Multiplication of Function: 𝒔 𝒙 = πŸπ’™πŸ βˆ’ 𝟐 +πŸ‘π’™πŸ‘ 𝒂𝒏𝒅 𝒕 𝒙 = βˆ’π’™ + 𝟏 Using the same given in example 2, find: (𝒔 βˆ— 𝒕) 𝟏 𝟐 𝒔 βˆ— 𝒕 𝒙 = βˆ’πŸ‘π’™πŸ’ +π’™πŸ‘ +πŸπ’™πŸ + πŸπ’™ βˆ’ 𝟐 = βˆ’πŸ‘ 𝟏 𝟐 πŸ’ + 𝟏 𝟐 πŸ‘ +𝟐 𝟏 𝟐 𝟐 + 𝟐 𝟏 𝟐 βˆ’ 𝟐
Multiplication of Function: = βˆ’πŸ‘ 𝟏 𝟐 πŸ’ + 𝟏 𝟐 πŸ‘ +𝟐 𝟏 𝟐 𝟐 + 𝟐 𝟏 𝟐 βˆ’ 𝟐 = βˆ’πŸ‘ 𝟏 πŸπŸ” + 𝟏 πŸ– + 𝟐 𝟏 πŸ’ + 𝟐 𝟏 𝟐 βˆ’ 𝟐 = βˆ’πŸ‘ πŸπŸ” + 𝟏 πŸ– + 𝟏 𝟐 + 𝟏 βˆ’ 𝟐 𝒔 βˆ— 𝒕 𝟏 𝟐 = βˆ’πŸ— πŸπŸ”
Multiplication of Function: 𝒔 βˆ— 𝒕 𝟏 𝟐 = βˆ’πŸ— πŸπŸ”
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.
Division of Function: Polynomial long- division Synthetic division
Example 1: 𝒇 𝒙 = πŸ‘π’™πŸ + πŸ’π’™ + πŸ“ 𝒂𝒏𝒅 π’ˆ 𝒙 = 𝒙 + 𝟐 πŸ‘π’™πŸ + πŸ’π’™ + πŸ“ 𝒙 + 𝟐 πŸ‘π’™ πŸ‘π’™πŸ βˆ’ πŸ”π’™ βˆ’ βˆ’πŸπ’™ +πŸ“ βˆ’πŸ βˆ’ πŸπ’™ +πŸ’ πŸ— 𝒇 π’ˆ 𝒙 = πŸ‘π’™ βˆ’ 𝟐 + πŸ— 𝒙 + 𝟐
Example: 𝒇 𝒙 = πŸ‘π’™πŸ + πŸ’π’™ + πŸ“ 𝒂𝒏𝒅 π’ˆ 𝒙 = 𝒙 + 𝟐 π‘Όπ’”π’Šπ’π’ˆ π‘Ίπ’šπ’π’•π’‰π’†π’•π’Šπ’„ π’…π’Šπ’—π’Šπ’”π’Šπ’π’: 𝒙 + 𝟐 = 𝟎 𝒙 + 𝟐 βˆ’ 𝟐 = 𝟎 βˆ’ 𝟐 βˆ’πŸ 𝒙 = βˆ’πŸ πŸ‘ πŸ’ πŸ“ πŸ‘ βˆ’πŸ” βˆ’πŸ πŸ’ πŸ— 𝒇 π’ˆ 𝒙 = πŸ‘π’™ βˆ’ 𝟐 + πŸ— 𝒙 + 𝟐 π’“π’†π’Žπ’‚π’Šπ’π’…π’†π’“
Division of Function: Using the same given in the example, find: 𝒇 𝒙 = πŸ‘π’™πŸ + πŸ’π’™ + πŸ“ 𝒂𝒏𝒅 π’ˆ 𝒙 = 𝒙 + 𝟐 𝒇 π’ˆ 𝟏 𝟐 𝒇 π’ˆ 𝒙 = πŸ‘π’™ βˆ’ 𝟐 + πŸ— 𝒙 + 𝟐 = πŸ‘ 𝟏 𝟐 βˆ’ 𝟐 + πŸ— = πŸ‘ 𝟐 + πŸ• 𝒇 π’ˆ 𝟏 𝟐 = πŸπŸ• 𝟐 π’π’“πŸ– 𝟏 𝟐
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.
Composite Function: f  g x f [gx] This is read β€œf composition g” or β€œf composed g” and means to copy the f function down but where ever you see an x, substitute in the g function.
Composite Function: Example 1: 𝒇 𝒙 = πŸ’π’™ + 𝟏𝟎 𝒂𝒏𝒅 π’ˆ 𝒙 = 𝒙 + 𝟏 = πŸ’(𝒙 + 𝟏) + 𝟏𝟎 πŸ’π’™ +πŸ’ +𝟏𝟎 πŸ’π’™ + πŸπŸ’ f [gx]=πŸ’π’™ + πŸπŸ’
Composite Function: Example 2: 𝒉 𝒙 = πŸ‘π’™πŸ βˆ’ 𝒙 + πŸ– 𝒂𝒏𝒅 π’Œ 𝒙 = βˆ’πŸπ’™ + πŸ‘ πŸ‘(βˆ’πŸπ’™ + πŸ‘ )𝟐 βˆ’ (βˆ’πŸπ’™ + πŸ‘) + πŸ– πŸ‘(πŸ’π’™πŸ βˆ’ πŸπŸπ’™ + πŸ—)+πŸπ’™ βˆ’ πŸ‘ +πŸ– πŸπŸπ’™πŸ βˆ’ πŸ‘πŸ”π’™ + πŸπŸ• + πŸπ’™ βˆ’ πŸ‘ + πŸ– β„Ž π‘˜ π‘₯ = 12π‘₯2 βˆ’ 34π‘₯ + 32
Another one… Given that: 𝒇 𝒙 = πŸ–π’™ + 𝟐 𝒂𝒏𝒅 π’ˆ 𝒙 = βˆ’πŸ‘π’™ βˆ’ πŸ•, π’‡π’Šπ’π’…: 𝟏. π’‡ο―π’ˆ 𝒙 2. π’‡ο―π’ˆ βˆ’πŸ πŸ–(βˆ’πŸ‘π’™ βˆ’ πŸ•) + 𝟐 βˆ’πŸπŸ’π’™ βˆ’ πŸ“πŸ” + 𝟐 f [gx]=βˆ’πŸπŸ’π’™βˆ’πŸ“πŸ’ βˆ’πŸπŸ’π’™ βˆ’ πŸ“πŸ’ βˆ’πŸπŸ’(βˆ’πŸ) βˆ’ πŸ“πŸ’ πŸ’πŸ– βˆ’ πŸ“πŸ’ 𝒇[π’ˆ βˆ’πŸ ]=βˆ’πŸ”
Quotation of the day: β€œ One of the lesson of math in our life that we should to apply is always be careful with the sign ” - Anonymous

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Operation of functions and Composite function.pptx

  • 2.
  • 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 = π’™πŸ βˆ’ πŸ‘ 𝒂𝒏𝒅 π’ˆ 𝒙 = πŸπ’™ + πŸ“ Example 1: βˆ’πŸ‘ +5 π’™πŸ βˆ’3 πŸπ’™ +πŸ“ + +𝟐 π’™πŸ +2x π’™πŸ + πŸπ’™ βˆ’ πŸ‘ + πŸ“ π’™πŸ + πŸπ’™ + 𝟐 𝑓 + 𝑔 π‘₯ = π‘₯2 + 2π‘₯ + 2
  • 5. Addition of Function: Example 2: 𝒂 𝒙 = π’™πŸ βˆ’ πŸ‘π’™ + 𝟏 βˆ’ πŸπ’™πŸ‘ 𝒂𝒏𝒅 𝒃 𝒙 = βˆ’πŸ“π’™ + π’™πŸ‘ βˆ’ 𝟏𝟎 𝒂 𝒙 = βˆ’πŸπ’™πŸ‘ + π’™πŸ βˆ’ πŸ‘π’™ + 𝟏 𝒃 𝒙 = π’™πŸ‘ βˆ’ πŸ“π’™ βˆ’ 𝟏𝟎 βˆ’πŸπ’™πŸ‘ + π’™πŸ βˆ’ πŸ‘π’™ + 𝟏 π’™πŸ‘ + βˆ’πŸ“π’™ βˆ’ 𝟏𝟎 βˆ’π’™πŸ‘ + π’™πŸ βˆ’πŸ–π’™ βˆ’πŸ— 𝒂 + 𝒃 𝒙 = βˆ’π’™πŸ‘ + π’™πŸ βˆ’ πŸ–π’™ βˆ’ πŸ—
  • 6. Addition of Function: Using the same given in example 2, find: 𝒂 𝒙 = π’™πŸ βˆ’ πŸ‘π’™ + 𝟏 βˆ’ πŸπ’™πŸ‘ 𝒂𝒏𝒅 𝒃 𝒙 = βˆ’πŸ“π’™ + π’™πŸ‘ βˆ’ 𝟏𝟎 (𝒂 + 𝒃)(βˆ’πŸ‘) 𝒂 + 𝒃 βˆ’πŸ‘ = πŸ“πŸ 𝒂 + 𝒃 𝒙 = βˆ’π’™πŸ‘ + π’™πŸ βˆ’ πŸ–π’™ βˆ’ πŸ— = βˆ’ βˆ’πŸ‘ πŸ‘ + βˆ’πŸ‘ 𝟐 βˆ’ πŸ–(βˆ’πŸ‘) βˆ’ πŸ— = βˆ’ βˆ’πŸπŸ• + πŸ— + πŸπŸ’ βˆ’ πŸ— = πŸπŸ• + πŸ— + πŸπŸ’ βˆ’ πŸ—
  • 7. Subtraction of Function:  fο€­ 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 = πŸ‘π’™πŸ + 𝟏𝟎 𝒂𝒏𝒅 π’ˆ 𝒙 = πŸ’π’™ + πŸ• πŸ‘π’™πŸ + 𝟏𝟎 (πŸ’π’™ + πŸ•) βˆ’ πŸ‘π’™πŸ + 𝟏𝟎 βˆ’πŸ’π’™ βˆ’πŸ• πŸ‘π’™πŸ βˆ’ πŸ’π’™ + 𝟏𝟎 βˆ’ πŸ• πŸ‘π’™πŸ βˆ’ πŸ’π’™ + πŸ‘ 𝒇 βˆ’ π’ˆ 𝒙 = πŸ‘π’™πŸ βˆ’ πŸ’π’™ + πŸ‘
  • 9. Subtraction of Function: Example 2: π’ˆ 𝒙 = βˆ’π’™πŸ + πŸ— βˆ’πŸπ’™πŸ‘ 𝒂𝒏𝒅 𝒉 𝒙 = βˆ’πŸ‘π’™ + πŸ’ βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ— (βˆ’πŸ‘π’™ + πŸ’) βˆ’ +πŸ‘π’™ βˆ’πŸ’ βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ‘π’™ + πŸ— βˆ’ πŸ’ π’ˆ βˆ’ 𝒉 𝒙 = βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ‘π’™ + πŸ“ βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ— βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ‘π’™ + πŸ“
  • 10. Subtraction of Function: π’ˆ 𝒙 = βˆ’π’™πŸ + πŸ— βˆ’πŸπ’™πŸ‘ 𝒂𝒏𝒅 𝒉 𝒙 = βˆ’πŸ‘π’™ + πŸ’ Using the same given in example 2, find: (π’ˆ βˆ’ 𝒉)(𝟐) π’ˆ βˆ’ 𝒉 𝒙 = βˆ’πŸπ’™πŸ‘ βˆ’π’™πŸ +πŸ‘π’™ + πŸ“ = βˆ’πŸ(𝟐)πŸ‘ βˆ’(𝟐)𝟐 +πŸ‘ 𝟐 + πŸ“ = βˆ’πŸ πŸ– βˆ’ (πŸ’) + πŸ” + πŸ“ = βˆ’πŸπŸ” βˆ’ πŸ’ + 𝟏𝟏 π’ˆ βˆ’ 𝒉 𝟐 = βˆ’πŸ—
  • 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 = βˆ’πŸ•π’™ + 𝟏𝟐 𝒂𝒏𝒅 π’ˆ 𝒙 = πŸπ’™ βˆ’ πŸ“ Example 1: βˆ’πŸ•π’™ + 𝟏𝟐 πŸπ’™ βˆ’ πŸ“ βˆ— βˆ’πŸπŸ’π’™πŸ + πŸπŸ’π’™ πŸ‘πŸ“π’™ βˆ’ πŸ”πŸŽ βˆ’πŸπŸ’π’™πŸ +πŸ“πŸ—π’™ βˆ’ πŸ”πŸŽ (βˆ’πŸ•π’™ + 𝟏𝟐)(πŸπ’™ βˆ’ πŸ“) 𝑭 = 𝑢 = 𝑰 = 𝑳 = βˆ’πŸπŸ’π’™πŸ πŸ‘πŸ“π’™ πŸπŸ’π’™ βˆ’πŸ”πŸŽ πŸ“πŸ—π’™ 𝒇 βˆ— π’ˆ 𝒙 = βˆ’πŸπŸ’π’™πŸ + πŸ“πŸ—π’™ βˆ’ πŸ”πŸŽ
  • 13. Multiplication of Function: Example 2: 𝒔 𝒙 = πŸπ’™πŸ βˆ’ 𝟐 +πŸ‘π’™πŸ‘ 𝒂𝒏𝒅 𝒕 𝒙 = βˆ’π’™ + 𝟏 πŸ‘π’™πŸ‘ +πŸπ’™πŸ βˆ’πŸ βˆ’π’™ + 𝟏 βˆ’πŸ‘π’™πŸ’ βˆ’πŸπ’™πŸ‘ +πŸπ’™ πŸ‘π’™πŸ‘ +πŸπ’™πŸ βˆ’πŸ βˆ’πŸ‘π’™πŸ’ +π’™πŸ‘ +πŸπ’™πŸ + πŸπ’™ βˆ’ 𝟐 𝒔 βˆ— 𝒕 𝒙 = βˆ’πŸ‘π’™πŸ’ +π’™πŸ‘ +πŸπ’™πŸ + πŸπ’™ βˆ’ 𝟐
  • 14. Multiplication of Function: 𝒔 𝒙 = πŸπ’™πŸ βˆ’ 𝟐 +πŸ‘π’™πŸ‘ 𝒂𝒏𝒅 𝒕 𝒙 = βˆ’π’™ + 𝟏 Using the same given in example 2, find: (𝒔 βˆ— 𝒕) 𝟏 𝟐 𝒔 βˆ— 𝒕 𝒙 = βˆ’πŸ‘π’™πŸ’ +π’™πŸ‘ +πŸπ’™πŸ + πŸπ’™ βˆ’ 𝟐 = βˆ’πŸ‘ 𝟏 𝟐 πŸ’ + 𝟏 𝟐 πŸ‘ +𝟐 𝟏 𝟐 𝟐 + 𝟐 𝟏 𝟐 βˆ’ 𝟐
  • 15. Multiplication of Function: = βˆ’πŸ‘ 𝟏 𝟐 πŸ’ + 𝟏 𝟐 πŸ‘ +𝟐 𝟏 𝟐 𝟐 + 𝟐 𝟏 𝟐 βˆ’ 𝟐 = βˆ’πŸ‘ 𝟏 πŸπŸ” + 𝟏 πŸ– + 𝟐 𝟏 πŸ’ + 𝟐 𝟏 𝟐 βˆ’ 𝟐 = βˆ’πŸ‘ πŸπŸ” + 𝟏 πŸ– + 𝟏 𝟐 + 𝟏 βˆ’ 𝟐 𝒔 βˆ— 𝒕 𝟏 𝟐 = βˆ’πŸ— πŸπŸ”
  • 16. Multiplication of Function: 𝒔 βˆ— 𝒕 𝟏 𝟐 = βˆ’πŸ— πŸπŸ”
  • 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 = πŸ‘π’™πŸ + πŸ’π’™ + πŸ“ 𝒂𝒏𝒅 π’ˆ 𝒙 = 𝒙 + 𝟐 πŸ‘π’™πŸ + πŸ’π’™ + πŸ“ 𝒙 + 𝟐 πŸ‘π’™ πŸ‘π’™πŸ βˆ’ πŸ”π’™ βˆ’ βˆ’πŸπ’™ +πŸ“ βˆ’πŸ βˆ’ πŸπ’™ +πŸ’ πŸ— 𝒇 π’ˆ 𝒙 = πŸ‘π’™ βˆ’ 𝟐 + πŸ— 𝒙 + 𝟐
  • 20. Example: 𝒇 𝒙 = πŸ‘π’™πŸ + πŸ’π’™ + πŸ“ 𝒂𝒏𝒅 π’ˆ 𝒙 = 𝒙 + 𝟐 π‘Όπ’”π’Šπ’π’ˆ π‘Ίπ’šπ’π’•π’‰π’†π’•π’Šπ’„ π’…π’Šπ’—π’Šπ’”π’Šπ’π’: 𝒙 + 𝟐 = 𝟎 𝒙 + 𝟐 βˆ’ 𝟐 = 𝟎 βˆ’ 𝟐 βˆ’πŸ 𝒙 = βˆ’πŸ πŸ‘ πŸ’ πŸ“ πŸ‘ βˆ’πŸ” βˆ’πŸ πŸ’ πŸ— 𝒇 π’ˆ 𝒙 = πŸ‘π’™ βˆ’ 𝟐 + πŸ— 𝒙 + 𝟐 π’“π’†π’Žπ’‚π’Šπ’π’…π’†π’“
  • 21. Division of Function: Using the same given in the example, find: 𝒇 𝒙 = πŸ‘π’™πŸ + πŸ’π’™ + πŸ“ 𝒂𝒏𝒅 π’ˆ 𝒙 = 𝒙 + 𝟐 𝒇 π’ˆ 𝟏 𝟐 𝒇 π’ˆ 𝒙 = πŸ‘π’™ βˆ’ 𝟐 + πŸ— 𝒙 + 𝟐 = πŸ‘ 𝟏 𝟐 βˆ’ 𝟐 + πŸ— = πŸ‘ 𝟐 + πŸ• 𝒇 π’ˆ 𝟏 𝟐 = πŸπŸ• 𝟐 π’π’“πŸ– 𝟏 𝟐
  • 22.
  • 23.
  • 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 β€œf composition g” or β€œf 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 = πŸ’π’™ + 𝟏𝟎 𝒂𝒏𝒅 π’ˆ 𝒙 = 𝒙 + 𝟏 = πŸ’(𝒙 + 𝟏) + 𝟏𝟎 πŸ’π’™ +πŸ’ +𝟏𝟎 πŸ’π’™ + πŸπŸ’ f [gx]=πŸ’π’™ + πŸπŸ’
  • 27. Composite Function: Example 2: 𝒉 𝒙 = πŸ‘π’™πŸ βˆ’ 𝒙 + πŸ– 𝒂𝒏𝒅 π’Œ 𝒙 = βˆ’πŸπ’™ + πŸ‘ πŸ‘(βˆ’πŸπ’™ + πŸ‘ )𝟐 βˆ’ (βˆ’πŸπ’™ + πŸ‘) + πŸ– πŸ‘(πŸ’π’™πŸ βˆ’ πŸπŸπ’™ + πŸ—)+πŸπ’™ βˆ’ πŸ‘ +πŸ– πŸπŸπ’™πŸ βˆ’ πŸ‘πŸ”π’™ + πŸπŸ• + πŸπ’™ βˆ’ πŸ‘ + πŸ– β„Ž π‘˜ π‘₯ = 12π‘₯2 βˆ’ 34π‘₯ + 32
  • 28. Another one… Given that: 𝒇 𝒙 = πŸ–π’™ + 𝟐 𝒂𝒏𝒅 π’ˆ 𝒙 = βˆ’πŸ‘π’™ βˆ’ πŸ•, π’‡π’Šπ’π’…: 𝟏. π’‡ο―π’ˆ 𝒙 2. π’‡ο―π’ˆ βˆ’πŸ πŸ–(βˆ’πŸ‘π’™ βˆ’ πŸ•) + 𝟐 βˆ’πŸπŸ’π’™ βˆ’ πŸ“πŸ” + 𝟐 f [gx]=βˆ’πŸπŸ’π’™βˆ’πŸ“πŸ’ βˆ’πŸπŸ’π’™ βˆ’ πŸ“πŸ’ βˆ’πŸπŸ’(βˆ’πŸ) βˆ’ πŸ“πŸ’ πŸ’πŸ– βˆ’ πŸ“πŸ’ 𝒇[π’ˆ βˆ’πŸ ]=βˆ’πŸ”
  • 29. Quotation of the day: β€œ One of the lesson of math in our life that we should to apply is always be careful with the sign ” - Anonymous