1. Section 5.3 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net
Inverse Functions
Section 5.3
Calculus AP/Dual, Revised ©2017
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§5.3: Derivative of Inverse Functions

2. §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
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§5.3: Derivative of Inverse Functions

3. §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
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§5.3: Derivative of Inverse Functions

4. §5.3: Derivative of Inverse Functions
Review Example 1
inverse
Mirrored Image
𝒚=𝒙
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§5.3: Derivative of Inverse Functions

5. 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
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§5.3: Derivative of Inverse Functions

6. 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
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§5.3: Derivative of Inverse Functions

7. §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
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§5.3: Derivative of Inverse Functions

8. Further Explanation of Inverses
The domain of 𝒇 is ______________ of 𝒈
The range of 𝒇 is ______________ of 𝒈
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§5.3: Derivative of Inverse Functions

9. §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?
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§5.3: Derivative of Inverse Functions

10. §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?
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§5.3: Derivative of Inverse Functions

11. §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.
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§5.3: Derivative of Inverse Functions

12. 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.
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§5.3: Derivative of Inverse Functions

13. §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 .
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§5.3: Derivative of Inverse Functions

14. The Derivative of the Inverse
Solve for 𝒈′ 𝒙 from 𝒇 𝒈 𝒙 =𝒙
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§5.3: Derivative of Inverse Functions

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

16. 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
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§5.3: Derivative of Inverse Functions

17. §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).
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§5.3: Derivative of Inverse Functions

18. §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).
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§5.3: Derivative of Inverse Functions

19. §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).
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§5.3: Derivative of Inverse Functions

20. §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 .
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§5.3: Derivative of Inverse Functions

21. §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 .
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§5.3: Derivative of Inverse Functions

22. §5.3: Derivative of Inverse Functions
Example 3
If 𝒇 𝒙 = 𝒙 𝟑 +𝟒. Solve for 𝒈′ 𝟓 where 𝒈 𝒙 = 𝒇 −𝟏 𝒙 .
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§5.3: Derivative of Inverse Functions

23. §5.3: Derivative of Inverse Functions
Your Turn
If 𝒇 𝒙 = 𝒙−𝟒 . Solve 𝒇 −𝟏 ′ 𝟓 .
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§5.3: Derivative of Inverse Functions

24. §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 .
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§5.3: Derivative of Inverse Functions

25. §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 .
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§5.3: Derivative of Inverse Functions

26. §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
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§5.3: Derivative of Inverse Functions

27. §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
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§5.3: Derivative of Inverse Functions

28. §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
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§5.3: Derivative of Inverse Functions

29. 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
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§5.3: Derivative of Inverse Functions

30. 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
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§5.3: Derivative of Inverse Functions

31. §5.3: Derivative of Inverse Functions
Assignment
Worksheet
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§5.3: Derivative of Inverse Functions