Section 5.3 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net Inverse Functions Section 5.3 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Review The result of exchanging the input and output value of a relation is an Inverse Function An inverse “undoes” the function. It switches (𝒙, 𝒚) to (𝒚, 𝒙) Interchange the 𝒙 and the 𝒚. (make 𝒚 𝒙 and make 𝒙 𝒚) Resolve for 𝒚. Written in function notation as 𝒇 −𝟏 (𝒙) Vertical Line Test: 𝒚 is a function of 𝒙 if and only if no vertical line intersects the graph at more than one point Horizontal Line Test: to see if it has an inverse “one to one” must pass the test to have an inverse 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Review Example 1 Determine the inverse of this relation, {(𝟎, −𝟑), (𝟐, 𝟏), 𝐚𝐧𝐝 (𝟔, 𝟑)} …to Solve the inverse, switch the x’s and y’s {(𝟎, −𝟑), (𝟐, 𝟏), (𝟔, 𝟑)} {( , ), ( , ), ( , )} –3 2 1 6 3 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Review Example 1 inverse Mirrored Image 𝒚=𝒙 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

Question If a function is a relation, is an inverse a function as well? inverse NO REPEATING 𝒀’𝒔 To Solve 𝒇′(𝒙), we have the HORIZONTAL LINE TEST 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

y = 3x – 2 y = 3 – 2 x x = 3y – 2 x + 2 = 3y Review Example 2 Determine the inverse of 𝒚=𝟑𝒙−𝟐 …to Solve the inverse, switch the 𝒙’s and 𝒚’s y = 3x – 2 y = 3 – 2 x x = 3y – 2 x + 2 = 3y 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Review If 𝒈 has an inverse of 𝒇, then 𝒇 is the inverse of 𝒈 A function has an inverses if: One–to–One Function (passes the horizontal line test) If 𝒇 is strictly “monotonic” (strictly increasing or decreasing over the entire interval) or one–way 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

Further Explanation of Inverses The domain of 𝒇 is ______________ of 𝒈 The range of 𝒇 is ______________ of 𝒈 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Review Example 3 Is this graph a function? Does it have an inverse? Is it one–to–one function and monotonic? 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Review Example 4 Is this graph a function? Does it have an inverse? Is it one–to–one function and monotonic? 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Review Example 5 Prove that 𝒇 𝒙 =𝟐𝒙−𝟔 and 𝒇 −𝟏 𝒙 = 𝟏 𝟐 𝒙+𝟑 are inverses through a composition. 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

The Derivative of an Inverse Function Let 𝒇 be a function that is differentiable on an interval 𝑰. If 𝒇 has an inverse function 𝒈, then 𝒈 is differentiable at any 𝒙 for which 𝒇 −𝟏 ′ (𝒂)= 𝟏 𝒇 ′ 𝒇 −𝟏 𝒂 OR 𝒈′ 𝒙 = 𝟏 𝒇 ′ 𝒈 𝒙 Slopes of the original functions and its derivative are reciprocal slopes of each other. 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Steps State the given by listing as points Make the equation equal to the inverse’s derivative, equate to zero, and solve for 𝒙. Take the derivative of the original function and plug in what is given. Take the reciprocal of the answer to establish the 𝒇 −𝟏 . 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

The Derivative of the Inverse Solve for 𝒈′ 𝒙 from 𝒇 𝒈 𝒙 =𝒙 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

The Derivative of the Inverse Therefore, 𝒈′ 𝒙 = 𝒇 −𝟏 𝒙 Original 𝒙–value on 𝒇 [INPUT] Original 𝒚–value on 𝒇 [OUTPUT] 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

Proving the Inverse Point’s Slope Blue: 𝒇 𝒙 =𝒙𝟑 Red: 𝒈 𝒙 = 𝒙 𝟏/𝟑 𝟐, 𝟖 𝒇 𝒙 =𝒙𝟑 𝒇 ′ 𝒙 =𝟑𝒙𝟐 Point Slope 𝟏, 𝟏 𝟐, 𝟖 𝟑, 𝟐𝟕 𝒈 𝒙 = 𝒙 𝟏/𝟑 𝒈 ′ 𝒙 = 𝟏 𝟑 𝒙 −𝟐/𝟑 Point Slope 𝟏, 𝟏 𝟖, 𝟐 𝟐𝟕, 𝟑 (𝟖, 𝟐) Slope 𝟑 𝟏𝟐 𝟐𝟕 Slope 𝟏/𝟑 𝟏/𝟏𝟐 𝟏/𝟐𝟕 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Example 1 Let 𝒇 be a function defined by 𝒇 𝒙 = 𝒙 𝟐 for which 𝒙≥𝟎, 𝒈 𝒙 = 𝒇 −𝟏 𝒙 and 𝒈 𝟒 =𝟐. Solve for 𝒈′(𝟒). 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Example 1 Let 𝒇 be a function defined by 𝒇 𝒙 = 𝒙 𝟐 for which 𝒙≥𝟎, 𝒈 𝒙 = 𝒇 −𝟏 𝒙 and 𝒈 𝟒 =𝟐. Solve for 𝒈′(𝟒). 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Example 1 Let 𝒇 be a function defined by 𝒇 𝒙 = 𝒙 𝟐 for which 𝒙≥𝟎, 𝒈 𝒙 = 𝒇 −𝟏 𝒙 and 𝒈 𝟒 =𝟐. Solve for 𝒈′(𝟒). 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Example 2 Let 𝒇 be a function defined by 𝒇 𝒙 = 𝒙 𝟑 +𝒙, 𝒈 𝒙 = 𝒇 −𝟏 𝒙 and 𝒈 𝟐 =𝟏. Solve for 𝒈 ′ 𝟐 . 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Example 2 Let 𝒇 be a function defined by 𝒇 𝒙 = 𝒙 𝟑 +𝒙, 𝒈 𝒙 = 𝒇 −𝟏 𝒙 and 𝒈 𝟐 =𝟏. Solve for 𝒈 ′ 𝟐 . 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Example 3 If 𝒇 𝒙 = 𝒙 𝟑 +𝟒. Solve for 𝒈′ 𝟓 where 𝒈 𝒙 = 𝒇 −𝟏 𝒙 . 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Your Turn If 𝒇 𝒙 = 𝒙−𝟒 . Solve 𝒇 −𝟏 ′ 𝟓 . 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Example 4 Given 𝒇 𝟑 =𝟓, 𝒇 ′ 𝟑 =𝟕, 𝒇 𝟐 =𝟑, and 𝒇 ′ 𝟐 =−𝟒 and 𝒇 and 𝒈 are inverses, solve for 𝒈 ′ 𝟑 . 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Your Turn Given 𝒇 𝟒 =𝟓, 𝒇 ′ 𝟒 =𝟑, 𝒇 𝟓 =𝟒, and 𝒇 ′ 𝟓 =−𝟐 and 𝒇 and 𝒈 are inverses, solve for 𝒈 ′ 𝟓 . 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Example 5 Given the table below. Solve for 𝒈 ′ 𝟐 and 𝒇 and 𝒈 are inverses. 𝒙 𝟐 𝟒 𝟔 𝟖 𝟏𝟎 𝒇 𝒙 𝟏 𝟎 𝒇 ′ 𝒙 −𝟏 𝟑 𝟏 𝟐 𝟓 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Example 6 Given the table below. Solve for 𝒈 ′ 𝟔 and 𝒇 and 𝒈 are inverses. 𝒙 𝟐 𝟒 𝟔 𝟖 𝟏𝟎 𝒇 𝒙 𝟏 𝟎 𝒇 ′ 𝒙 −𝟏 𝟑 𝟏 𝟐 𝟓 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Your Turn Given the table below. Solve for 𝒈 ′ 𝟑 and 𝒇 and 𝒈 are inverses. 𝒙 −𝟏 𝟎 𝟏 𝟐 𝟑 𝒇 𝒙 𝒇 ′ 𝒙 −𝟐 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

AP Multiple Choice Practice Question 1 (Non-Calculator) Given the table below. Solve for 𝒈 ′ 𝟏 and 𝒇 and 𝒈 are inverses. (A) 𝟏 (B) 𝟏/𝟒 (C) 𝟒 (D) 𝟏/𝟑 𝒙 𝟑 𝟒 𝟕 𝟗 𝒇 𝒙 𝟏 𝟖 𝒇 ′ 𝒙 𝟏 𝟒 𝟑 𝟕 𝟐 𝟕 𝟖 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

AP Multiple Choice Practice Question 1 (Non-Calculator) Given the table below. Solve for 𝒈 ′ 𝟏 and 𝒇 and 𝒈 are inverses. 𝒙 𝟑 𝟒 𝟕 𝟗 𝒇 𝒙 𝟏 𝟖 𝒇 ′ 𝒙 𝟏 𝟒 𝟑 𝟕 𝟐 𝟕 𝟖 Vocabulary Process and Connections Answer and Justifications 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions

§5.3: Derivative of Inverse Functions Assignment Worksheet 4/15/2019 5:11 PM §5.3: Derivative of Inverse Functions