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)
[table][br][tr][br][td][b]Factual Questions[/b][/td][br][td][b]Conceptual Questions[/b][/td][br][td][b]Debatable Questions[/b][/td][br][/tr][br][tr][br][td]1. What is the definition of an inverse function?[/td][br][td]1. Why does a function need to be bijective (one-to-one and onto) to have an inverse function?[/td][br][td]1. Is finding the inverse of a function more challenging than finding the function itself? Why or why not?[/td][br][/tr][br][tr][br][td]2. How do you find the inverse of the function[math]f(x)=2x+3[/math]?[/td][br][td]2. Discuss the graphical relationship between a function and its inverse.[/td][br][td]2. Can the concept of inverse functions be applied to solve real-world problems effectively?[/td][br][/tr][br][tr][br][td]3. What condition must a function meet to have an inverse that is also a function?[/td][br][td]3. Explain the importance of the horizontal line test in determining if a function has an inverse.[/td][br][td]3. Debate the significance of understanding inverse functions in the curriculum.[/td][br][/tr][br][tr][br][td]4. Determine the inverse of [math]f(x)=\frac{1}{x-1}[/math].[/td][br][td]4. How does the composition of a function and its inverse relate to the identity function?[/td][br][td]4. Discuss the statement: "The ability to find and use inverse functions is crucial for understanding advanced mathematics."[/td][br][/tr][br][tr][br][td]5. Explain how to verify that two functions are inverses of each other.[/td][br][td]5. Compare the process of finding inverses for linear and non-linear functions.[/td][br][td]5. Evaluate the impact of technology on teaching and learning about inverse functions.[/td][br][/tr][br][/table][br]
[b]Investigating the geometric relationship between a function and its inverse.[/b][br][br]Experiment with the sliders for [math]a,b[/math] to change the original function. Consider what you observe.
1. Reflection and Symmetry:[br] - How does the graph of the inverse function relate to the graph of the original function in terms of reflection and symmetry?[br] - Can you find a line over which both the function and its inverse are symmetrical?[br][br]
2. Intercepts:[br] - What happens to the y-intercept of the original function when you find its inverse?[br] - How does the x-intercept of the function relate to the x-intercept of the inverse function?[br][br]
3. Intersection Points:[br] - Do the function and its inverse always intersect? If so, where is that point located in relation to the line [math]y=x[/math]?[br] - What is the significance of the point where the function and its inverse intersect?[br][br]
4. Function Behavior:[br] - How does the slope of the original function affect the slope of the inverse function?[br] - What does the intersection of a function and its inverse tell you about the behavior of the function?[br]
5. Finding Inverses:[br] - How can you determine the inverse of a given function just by looking at the graph?[br] - What changes in the graph when you alter the function's equation?
6. Function Types:[br] - Are there functions that do not have an inverse that is also a function? How can you identify them on the graph?[br] - What characteristics do functions with graphable inverses share?
7. Exploration with Changes:[br] - What happens to the inverse if you change the slope or intercept of the original function?[br] - How can you predict the graph of an inverse function based on the graph of the original function?
Part 2 - Finding the inverse algebraically
Watch the video below
If [math]g(x)=3x-4[/math], what is the inverse function [math]g^{-1}(x)[/math]?[br]
If [math]k(x)=sin(x)[/math], which of the following represents the inverse function [math]k^{-1}(x)[/math]?[br]
Given [math]j(x)=log(x)[/math], what is the inverse function [math]j^{-1}(x)[/math]?[br][br]
For the function [math]h(x)=1/(x+2)[/math], identify the inverse function [math]h^{-1}(x)[/math].
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
[MAA 2.4-2.5] COMPOSITION - INVERSE FUNCTION
[MAA 2.4-2.5] COMPOSITION - INVERSE FUNCTION_solutions
Lesson Plan- Exploring Inverse Functions
Inverse functions- Intuition pump (thought experiments and analogies)
