Unlocking Inverse Functions: An Algebraic Approach

PreAlgebra Grades High School 2:17 Video
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Lesson Description

Explore the concept of inverse functions, their algebraic determination, and graphical representation through reflections over the line y = x.

Video Resource

Find the Inverse of a Function Algebraically

Mario's Math Tutoring

Duration: 2:17
Watch on YouTube

Key Concepts

  • Inverse function definition: A function that 'undoes' the original function.
  • Algebraic method for finding inverses: Switching x and y, then solving for y.
  • Graphical representation of inverses: Reflection over the line y = x.

Learning Objectives

  • Students will be able to algebraically determine the inverse of a given function.
  • Students will be able to explain the relationship between a function and its inverse graphically.
  • Students will be able to verify that a function is indeed the inverse of another function.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of a function and its input/output relationship. Introduce the idea of an 'inverse' as something that reverses this relationship.
  • Video Explanation (10 mins)
    Watch the video 'Find the Inverse of a Function Algebraically' by Mario's Math Tutoring. Pay close attention to the steps for finding the inverse and the graphical representation.
  • Algebraic Method (15 mins)
    Demonstrate the algebraic method step-by-step: Replace f(x) with y, swap x and y, solve for the new y, and replace the new y with f⁻¹(x). Work through multiple examples, including f(x) = 2x - 8 (from the video) and other linear functions.
  • Graphical Representation (10 mins)
    Explain that the graph of the inverse function is a reflection of the original function over the line y = x. Show examples graphically using Desmos or other graphing software. Discuss how to graph y = x.
  • Verification (5 mins)
    Explain how to verify if two functions are inverses by composing them. That is, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Interactive Exercises

  • Algebraic Practice
    Provide students with a worksheet containing various functions (linear, quadratic, and rational) and have them find the inverse algebraically. Walk around the classroom and provide support as needed.
  • Graphical Verification
    Using Desmos or other graphing software, have students graph a function and its inverse. Ask them to visually confirm that the graphs are reflections of each other over y = x.

Discussion Questions

  • What does it mean for a function to 'undo' another function?
  • How can you tell graphically if two functions are inverses of each other?
  • Why is it important to be able to find the inverse of a function?

Skills Developed

  • Algebraic manipulation
  • Graphical interpretation
  • Problem-solving

Multiple Choice Questions

Question 1:

What is the first step in finding the inverse of a function algebraically?

Correct Answer: Replace f(x) with y.

Question 2:

If f(x) = 3x + 2, what is f⁻¹(x)?

Correct Answer: (x - 2) / 3

Question 3:

The graph of an inverse function is a ________ of the original function over the line y = x.

Correct Answer: reflection

Question 4:

Which of the following is true about the composition of a function and its inverse?

Correct Answer: f(f⁻¹(x)) = x

Question 5:

Which function does not have an inverse?

Correct Answer: f(x) = x²

Question 6:

What is the domain restriction needed for f(x) = √(x-4) to have an inverse?

Correct Answer: x ≥ 4

Question 7:

If f(x) = x³ + 1, what is the inverse function, f⁻¹(x)?

Correct Answer: f⁻¹(x) = ³√(x - 1)

Question 8:

If a point (a, b) lies on the graph of f(x), what point must lie on the graph of f⁻¹(x)?

Correct Answer: (b, a)

Question 9:

Given f(x) = 5x - 3 and g(x) = (x + 3) / 5, what is the value of f(g(x))?

Correct Answer: x

Question 10:

For which type of function is the inverse found by reflecting over the line y = x?

Correct Answer: All functions

Fill in the Blank Questions

Question 1:

The inverse function is denoted as f ____(x).

Correct Answer: ⁻¹

Question 2:

To find the inverse algebraically, you first replace f(x) with ____.

Correct Answer: y

Question 3:

The graphs of f(x) and f⁻¹(x) are reflections over the line y = ____.

Correct Answer: x

Question 4:

If f(x) = x - 5, then f⁻¹(x) = x + ____.

Correct Answer: 5

Question 5:

The process of 'undoing' a function is known as finding the ____.

Correct Answer: inverse

Question 6:

For a function to have an inverse, it must pass the ______________ line test.

Correct Answer: horizontal

Question 7:

The composition of a function and its inverse results in ___________.

Correct Answer: x

Question 8:

If f(2) = 7, then for the inverse function, f⁻¹(7) = _______.

Correct Answer: 2

Question 9:

When finding the inverse, the x and y variables are ___________.

Correct Answer: switched

Question 10:

The range of a function becomes the _________ of its inverse.

Correct Answer: domain

Educational Standards

PC.F.1.5:Determine if a function has an inverse. Algebraically and graphically find the inverse or define any restrictions on the domain that meet the requirement for invertibility, and find the inverse on the restricted domain. PC.F.2.4:Verify by analytical methods that one function is the inverse of another. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features.

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