1. Operations with Functions
Section 1.8
Precalculus PreAP/Dual, Revised ©2018
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§1.8: Function Operations

2. Operations with Functions
To add/subtract functions, COMBINE LIKE TERMS
To multiply functions and divide functions, DISTRIBUTE properly with EXPONENTS
The domain for +/–/× will be the same but division may be different.
Add or Subtract: 𝒇 𝒙 ±𝒈 𝒙 or 𝒇±𝒈 𝒙
Multiply: 𝒇 𝒙 •𝒈 𝒙 or 𝒇 𝒈 𝒙
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§1.8: Function Operations
6.2 function operations

3. §1.8: Function Operations
Example 1
If given 𝒇 𝒙 =𝟒 𝒙 𝟏/𝟐 and 𝒈 𝒙 =𝟗 𝒙 𝟏/𝟐 , solve (𝒇+𝒈)(𝒙). Leave answer in exponential form and identify the domain.
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§1.8: Function Operations

4. §1.8: Function Operations
Example 2
If given 𝒇 𝒙 =𝒙𝟐+𝟐 and 𝒈 𝒙 =−𝟒𝒙+𝟕, solve 𝒇−𝒈 𝒙 and identify the domain.
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§1.8: Function Operations

5. §1.8: Function Operations
Example 3
If given 𝒇 𝒙 =𝟔𝒙 and 𝒈 𝒙 = 𝒙 𝟑/𝟒 , solve 𝒇𝒈 𝒙 . Leave answer in exponential form and identify the domain.
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§1.8: Function Operations

6. §1.8: Function Operations
Your Turn
If given 𝒇 𝒙 = 𝒙 𝟐 +𝟐 and 𝒈 𝒙 =−𝟒𝒙+𝟕, solve 𝒇 𝒈 𝒙 and identify the domain.
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§1.8: Function Operations

7. §1.8: Function Operations
Compositions
Determine what is substituted
Take the INSIDE function and replace it
Take the outside function and bring it down
Replace the variable with the leftover variable
Simplify the expression
Notation: They may give you 𝒇 𝒈 𝒙 or 𝒇∘𝒈 𝒙. The meaning is the same.
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§1.8: Function Operations

8. §1.8: Function Operations
Written Notation
It may be written as… It may not be written as:
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§1.8: Function Operations

9. §1.8: Function Operations
Example 4
If given 𝒇 𝒙 =𝟒𝒙 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒇 𝒈 𝒙
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§1.8: Function Operations

10. §1.8: Function Operations
Example 5
If given 𝒇(𝒙)=𝟒𝒙 and 𝒈(𝒙)=𝟐−𝒙, solve 𝒈∘𝒇 𝒙
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§1.8: Function Operations

11. §1.8: Function Operations
Example 6
If given 𝒇(𝒙)=𝟒𝒙 and 𝒈(𝒙)=𝟐−𝒙, solve 𝒈 𝒇 𝒇 𝒙
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§1.8: Function Operations

12. §1.8: Function Operations
Your Turn
If given 𝒇(𝒙)=𝟒𝒙 and 𝒈(𝒙)=𝟐−𝒙, solve 𝒈 𝒈 𝒙
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§1.8: Function Operations

13. §1.8: Function Operations
Example 7
If given 𝒇 𝒙 =𝟒 𝒙 −𝟏 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒈 𝒇 𝟐
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§1.8: Function Operations

14. §1.8: Function Operations
Example 8
If given 𝒇 𝒙 =𝟒 𝒙 −𝟏 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒇∘𝒈 𝟓
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§1.8: Function Operations

15. §1.8: Function Operations
Example 9
If given 𝒇 𝒙 =𝟒 𝒙 −𝟏 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒇 𝒈 𝒚
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§1.8: Function Operations

16. §1.8: Function Operations
Example 10
If given 𝒇 𝒙 =𝟒 𝒙 −𝟏 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒇∘𝒈 𝟒 +𝒇 𝟓
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§1.8: Function Operations

17. §1.8: Function Operations
Your Turn
If given 𝒇 𝒙 = 𝒙 𝟐 +𝟐𝒙−𝟏 and 𝒈 𝒙 = 𝟏 𝒙−𝟑 , solve 𝒈∘𝒇 (𝒙)
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§1.8: Function Operations

18. §1.8: Function Operations
Example 11
Using the graph below, evaluate 𝒇 𝒈 𝟎 .
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§1.8: Function Operations

19. §1.8: Function Operations
Your Turn
Using the graph below, evaluate 𝒈 𝒇 −𝟏 .
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§1.8: Function Operations

20. §1.8: Function Operations
Example 12
You have a $10 gift card to a paint store. The store is offering 15% off your entire purchase of any paints and painting supplies. You decide to purchase a $30 can of paint and $25 worth of painting supplies. You have the option to use the gift card or the discount first. Which one should you first? Justify your reasoning.
Total amount of paying for paint:
Function for the $10 gift card:
Function for the 15% discount:
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§1.8: Function Operations

21. §1.8: Function Operations
Example 12
You have a $10 gift card to a paint store. The store is offering 15% off your entire purchase of any paints and painting supplies. You decide to purchase a $30 can of paint and $25 worth of painting supplies. You have the option to use the gift card or the discount first. Which one should you first? Justify your reasoning.
Use the 15% discount first 𝒈(𝒙): Use the Gift Card first 𝒇(𝒙):
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§1.8: Function Operations

22. §1.8: Function Operations
Assignment
Page EOO, 31 and 33 (a to c), odd
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§1.8: Function Operations
6.2 function operations