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

Last updated over 4 years ago

29 Nsɛmmisa

Hyɛ no nsow a efi ɔkyerɛwfo no hɔ:

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.

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F.IF.1

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100

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

Last updated over 4 years ago

29 Nsɛmmisa

Hyɛ no nsow a efi ɔkyerɛwfo no hɔ:

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

If you decrease an amount x by 20%, you have 0.8x. If you then increase 0.8x by 20%, you get (0.8x)(1.2) = 0.96x, which is less than the original amount x. The headline states that the mayor's salary was restored, but the mayor's salary will be 4% less than the original salary. A more appropriate headline would be Most of Mayor's Salary Restored.

If you decrease an amount x by 20%, you have 0.2x. If you then increase 0.2x by 20%, you get (0.2x)(1.2) = 0.24x, which is less than the original amount x. The headline states that the mayor's salary was restored, but the mayor's salary will be 76% less than the original salary. A more appropriate headline would be Some of Mayor's Salary Restored.

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

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

t is a function; the inverse of t is not a function because there are two y-values for one x-value.

t is not a function; the inverse of t is also not a function; in both t and its inverse, there are two y-values for at least one x-value.

t is a function; the inverse of t is also a function: in both t and its inverse, there is only one y-value for each x-value.

t is a function; the inverse of t is also a function because there is only one y-value for each x-value.

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Problem 2 Got It?

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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.

Yɛayi Graphing asɛmmisa type foforo a wɔatu mpɔn adi! Asuafo rentumi mmua saa asɛmmisa yi bio.

F.BF.4.a

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Problem 4 Got It?

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Problem 4 Got It?

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Problem 4 Got It?

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Problem 4 Got It?

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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?

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

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Problem 6 Got It?

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Problem 6 Got It?

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

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

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

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

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F.IF.1

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

For the relation to be a function, x may not assume 2, 3, 4, 5, or 6.

For the relation to be a function, x may not assume 2 or 5.

For the relation to be a function, x may not assume 3, 4, or 5.

x may assume any real number and the relation will still represent a function.

F.IF.1

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.

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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.

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Review Lesson 6-1: Find each real root.

No real roots

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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.

Yɛayi Graphing asɛmmisa type foforo a wɔatu mpɔn adi! Asuafo rentumi mmua saa asɛmmisa yi bio.

F.IF.7.a

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 __?__.

F.IF.1

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.

Reflection: Math Success

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F.IF.1

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

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

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?

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

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.

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?

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

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.