Inverse functions

Keywords

역함수 (Inverse Function), 수평선 검사 (Horizontal Line Test), 항등 함수 (Identity Function), 선형 함수의 역 (Linear Function Inverse), 비선형 함수의 역 (Non-linear Function Inverse), 그래픽 관계 (Graphical Relationship), 반사 및 대칭 (Reflection and Symmetry), Y절편 (Y-intercept), X절편 (X-intercept), 기울기 (Slope), 함수의 합성 (Composition of Functions)

Factual Questions	Conceptual Questions	Debatable Questions
1. What is the definition of an inverse function?	1. Why does a function need to be bijective (one-to-one and onto) to have an inverse function?	1. Is finding the inverse of a function more challenging than finding the function itself? Why or why not?
2. How do you find the inverse of the function?	2. Discuss the graphical relationship between a function and its inverse.	2. Can the concept of inverse functions be applied to solve real-world problems effectively?
3. What condition must a function meet to have an inverse that is also a function?	3. Explain the importance of the horizontal line test in determining if a function has an inverse.	3. Debate the significance of understanding inverse functions in the curriculum.
4. Determine the inverse of .	4. How does the composition of a function and its inverse relate to the identity function?	4. Discuss the statement: "The ability to find and use inverse functions is crucial for understanding advanced mathematics."
5. Explain how to verify that two functions are inverses of each other.	5. Compare the process of finding inverses for linear and non-linear functions.	5. Evaluate the impact of technology on teaching and learning about inverse functions.

Investigating the geometric relationship between a function and its inverse. Experiment with the sliders for

to change the original function. Consider what you observe.

1. Reflection and Symmetry: - How does the graph of the inverse function relate to the graph of the original function in terms of reflection and symmetry? - Can you find a line over which both the function and its inverse are symmetrical?

2. Intercepts: - What happens to the y-intercept of the original function when you find its inverse? - How does the x-intercept of the function relate to the x-intercept of the inverse function?

3. Intersection Points: - Do the function and its inverse always intersect? If so, where is that point located in relation to the line ? - What is the significance of the point where the function and its inverse intersect?

4. Function Behavior: - How does the slope of the original function affect the slope of the inverse function? - What does the intersection of a function and its inverse tell you about the behavior of the function?

5. Finding Inverses: - How can you determine the inverse of a given function just by looking at the graph? - What changes in the graph when you alter the function's equation?

6. Function Types: - Are there functions that do not have an inverse that is also a function? How can you identify them on the graph? - What characteristics do functions with graphable inverses share?

7. Exploration with Changes: - What happens to the inverse if you change the slope or intercept of the original function? - How can you predict the graph of an inverse function based on the graph of the original function?

Part 2 - Finding the inverse algebraically

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If , what is the inverse function ?

Select all that apply

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A)
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B)
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D)

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If , which of the following represents the inverse function ?

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Given , what is the inverse function ?

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For the function , identify the inverse function .

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B
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D
None of the above

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Part 3 - Testing your understanding of concept

Standard inverse functions questions Q2,3,5,7,Difficult inverse functions questions. Q38,39,44Other questions require knowledge of composite functions before attempting

Inverse functions

Keywords

Part 2 - Finding the inverse algebraically

Part 3 - Testing your understanding of concept

[MAA 2.4-2.5] COMPOSITION - INVERSE FUNCTION

[MAA 2.4-2.5] COMPOSITION - INVERSE FUNCTION_solutions

Lesson Plan- Exploring Inverse Functions