Algebra 2 6-7 Complete Lesson: Inverse Relations and Functions

Last updated almost 4 years ago

Note from the author:

A complete formative lesson with embedded slideshow, mini lecture screencasts, checks for understanding, practice items, mixed review, and reflection. I create these assignments to supplement each lesson of Pearson's Common Core Edition Algebra 1, Algebra 2, and Geometry courses. See also mathquest.net and twitter.com/mathquestEDU.

Solve It! What's wrong with the headline? Why? What is a more appropriate headline?

Problem 1 Got It? What are the graphs of t and its inverse? Represent both relations as mapping diagrams on the canvas.

Problem 1 Got It? Reasoning: Is t a function? Is the inverse of t a function? Explain.

Problem 2 Got It?

Problem 3 Got It? Graph the function, its inverse, and the line y = x on the same plane.
Note that the function and its inverse should be reflective across the line y = x.
Zoom and pan your graph to establish an appropriate viewing window.

We have released a new and improved Graphing question type! Students will no longer be able to answer this question.

Problem 4 Got It?

Problem 5 Got It? The function d, below, relates the distance d, in meters, that an object has fallen to its velocity, v, in meters per second.
What is the inverse of this function?

If a cliff diver falls from a height of 24 meters, what is his velocity in meters per second as he enters the water?

Problem 6 Got It? Given g(x) below, find g-1(x).

Problem 6 Got It?

What is the inverse of f ? Is the inverse a function?

For h, find the value for x for which the equality
(h ◦ h-1)(x) = x does not hold.

Reasoning: A function consists of the pairs (2, 3), (x, 4), and (5, 6). What values, if any, may x not assume?

Error Analysis: A classmate says the following.
Show that this is incorrect by finding examples of f(x) and g(x) for which the equation does not hold.

Review Lesson 6-6: For the given functions f, g, and h, match each function operation and composition on the left with the equivalent simplified expression on the right.

Review Lesson 6-1: Find each real root.

No real roots

Review Lesson 4-1: Graph the functions on the same plane.
Note how the functions are transformations of the parent quadratic function y = x2.
Zoom and pan your graph to establish an appropriate viewing window.

We have released a new and improved Graphing question type! Students will no longer be able to answer this question.

Vocabulary Review: Fill in each blank on the right with the appropriate item from the left.

Consider the relation below for some items.

The domain of a relation is the set of inputs, also called the __?__ coordinates of the ordered pair.
The range of a relation is the set of inputs, also called the __?__ coordinates of the ordered pair.
The domain of the relation above is __?__.
The range of the relation above is __?__.

Use Your Vocabulary: Categorize the domains and ranges on the left.

Use the relation r, as defined here.

Domain of r
Range of r
Domain of the inverse of r
Range of the inverse of r

100

Notes: Take a clear picture or screenshot of your Cornell notes for this lesson. Upload it to the canvas. Zoom and pan as needed.

For a refresher on the Cornell note-taking system, click here.

Algebra 2 6-7 Complete Lesson: Inverse Relations and Functions

Solve It! What's wrong with the headline? Why? What is a more appropriate headline?

Problem 1 Got It? What are the graphs of t and its inverse? Represent both relations as mapping diagrams on the canvas.

Problem 1 Got It? Reasoning: Is t a function? Is the inverse of t a function? Explain.

Problem 2 Got It?

Problem 3 Got It? Graph the function, its inverse, and the line y = x on the same plane. Note that the function and its inverse should be reflective across the line y = x. Zoom and pan your graph to establish an appropriate viewing window.

Problem 4 Got It?

Problem 4 Got It?

Problem 4 Got It?

Problem 4 Got It?

Problem 5 Got It? The function d, below, relates the distance d, in meters, that an object has fallen to its velocity, v, in meters per second. What is the inverse of this function?If a cliff diver falls from a height of 24 meters, what is his velocity in meters per second as he enters the water?

Problem 6 Got It? Given g(x) below, find g-1(x).

Problem 6 Got It?

Problem 6 Got It?

What is the inverse of f ? Is the inverse a function?

What is the inverse of f ? Is the inverse a function?

What is the inverse of f ? Is the inverse a function?

For h, find the value for x for which the equality (h ◦ h-1)(x) = x does not hold.

Reasoning: A function consists of the pairs (2, 3), (x, 4), and (5, 6). What values, if any, may x not assume?

Error Analysis: A classmate says the following.Show that this is incorrect by finding examples of f(x) and g(x) for which the equation does not hold.

Review Lesson 6-6: For the given functions f, g, and h, match each function operation and composition on the left with the equivalent simplified expression on the right.

Review Lesson 6-1: Find each real root.

Review Lesson 4-1: Graph the functions on the same plane. Note how the functions are transformations of the parent quadratic function y = x2. Zoom and pan your graph to establish an appropriate viewing window.

Vocabulary Review: Fill in each blank on the right with the appropriate item from the left. Consider the relation below for some items.

Use Your Vocabulary: Categorize the domains and ranges on the left. Use the relation r, as defined here.

Notes: Take a clear picture or screenshot of your Cornell notes for this lesson. Upload it to the canvas. Zoom and pan as needed.For a refresher on the Cornell note-taking system, click here.

Reflection: Math Success

Problem 3 Got It? Graph the function, its inverse, and the line y = x on the same plane.
Note that the function and its inverse should be reflective across the line y = x.
Zoom and pan your graph to establish an appropriate viewing window.

Problem 5 Got It? The function d, below, relates the distance d, in meters, that an object has fallen to its velocity, v, in meters per second.
What is the inverse of this function?

If a cliff diver falls from a height of 24 meters, what is his velocity in meters per second as he enters the water?

For h, find the value for x for which the equality
(h ◦ h-1)(x) = x does not hold.

Error Analysis: A classmate says the following.
Show that this is incorrect by finding examples of f(x) and g(x) for which the equation does not hold.

Review Lesson 4-1: Graph the functions on the same plane.
Note how the functions are transformations of the parent quadratic function y = x2.
Zoom and pan your graph to establish an appropriate viewing window.

Vocabulary Review: Fill in each blank on the right with the appropriate item from the left.

Consider the relation below for some items.

Use Your Vocabulary: Categorize the domains and ranges on the left.

Use the relation r, as defined here.

Notes: Take a clear picture or screenshot of your Cornell notes for this lesson. Upload it to the canvas. Zoom and pan as needed.

For a refresher on the Cornell note-taking system, click here.