Operations with Functions Section 1.8 Precalculus PreAP/Dual, Revised ©2018 viet.j.dang@humbleisd.net 2/27/2019 9:04 PM §1.8: Function Operations

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 2/27/2019 9:04 PM §1.8: Function Operations 6.2 function operations

§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. 2/27/2019 9:04 PM §1.8: Function Operations

§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. 2/27/2019 9:04 PM §1.8: Function Operations

§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. 2/27/2019 9:04 PM §1.8: Function Operations

§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. 2/27/2019 9:04 PM §1.8: Function Operations

§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. 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Written Notation It may be written as… It may not be written as: 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Example 4 If given 𝒇 𝒙 =𝟒𝒙 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒇 𝒈 𝒙 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Example 5 If given 𝒇(𝒙)=𝟒𝒙 and 𝒈(𝒙)=𝟐−𝒙, solve 𝒈∘𝒇 𝒙 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Example 6 If given 𝒇(𝒙)=𝟒𝒙 and 𝒈(𝒙)=𝟐−𝒙, solve 𝒈 𝒇 𝒇 𝒙 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Your Turn If given 𝒇(𝒙)=𝟒𝒙 and 𝒈(𝒙)=𝟐−𝒙, solve 𝒈 𝒈 𝒙 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Example 7 If given 𝒇 𝒙 =𝟒 𝒙 −𝟏 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒈 𝒇 𝟐 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Example 8 If given 𝒇 𝒙 =𝟒 𝒙 −𝟏 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒇∘𝒈 𝟓 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Example 9 If given 𝒇 𝒙 =𝟒 𝒙 −𝟏 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒇 𝒈 𝒚 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Example 10 If given 𝒇 𝒙 =𝟒 𝒙 −𝟏 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒇∘𝒈 𝟒 +𝒇 𝟓 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Your Turn If given 𝒇 𝒙 = 𝒙 𝟐 +𝟐𝒙−𝟏 and 𝒈 𝒙 = 𝟏 𝒙−𝟑 , solve 𝒈∘𝒇 (𝒙) 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Example 11 Using the graph below, evaluate 𝒇 𝒈 𝟎 . 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Your Turn Using the graph below, evaluate 𝒈 𝒇 −𝟏 . 2/27/2019 9:04 PM §1.8: Function Operations

§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: 2/27/2019 9:04 PM §1.8: Function Operations

§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): 2/27/2019 9:04 PM §1.8: Function Operations

§1.8: Function Operations Assignment Page 81 3-23 EOO, 31 and 33 (a to c), 35-45 odd 2/27/2019 9:04 PM §1.8: Function Operations 6.2 function operations