Alg2 Lesson 2.4: Inverse Functions

Lesson 2.4: Inverse

Functions

Obj: I can find the inverse of functions.

EQ: How do I find the inverse of a linear function?
How do I find inverse functions given a table and
graph?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math
tools to communicate your mathematics, reasons, and
connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may
arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Poll

What do you plan to do right after these lesson slides?

• Highlight and paraphrase notes in journal

• Deltamath

• Exit Ticket on Canvas

• Take a nap

Part 1: Using Inverses in

Context

Work with a Partner to Fill in the Table

Eq into Words

Equation

Inverse Actions of Words

½x - 5 = 12

3(x - 4)^2 = 15

5^x = 40

In general, what are the inverse operations doing?

Dropdown

The inverse operations are ​

Select answer

the operations that were ​

Select answer

in each equation, but in ​

Select answer

order.

Handout 2.4.A: Making a Tiny Home

Your guardians plan to move into a tiny home after you move out of their
house. Their research indicates that tiny homes on mobile trailers are
often no more than 8 feet (ft) wide to avoid requiring a wide-load permit.
Additionally, tiny homes can be at most 13.5 ft tall to pass under highway
bridges. To prevent any accidents in transit, your guardians decide to limit
the height of their tiny home to 13 ft. The length of a tiny home can vary,
although typical trailer lengths are usually between 8 ft and 30 ft.

Multiple Select

What do we know about the tiny home?

• maximum possible measures of some of the dimensions

• minimum possible measures of some of the dimensions

• width

• height

• length

Drag and Drop

What are we trying to determine in the first problem? We are trying to write a formula for the ​

Placeholder

of the siding. We need to determine the lateral (​

Placeholder

) ​

Placeholder

of the home.

Drag these tiles and drop them in the correct blank above

area

total side

volume

perimeter

base

Multiple Choice

What is the most common shape of tiny homes (and other buildings present day)?

• Triangular shape
• Circular shape
• L-shaped design
• Rectangular or box-like shape

Dropdown

What assumptions can we make so that our calculations are as simple as possible? If we assume the tiny home has no windows and no doors, then our calculations are simple but we will have ​

Select answer

the area.

Drag and Drop

We know that C was a function. What about this new reversed relationship, is it a function? What should be the input and what should be the output? If we consider the ​

Placeholder

as the input and the ​

Placeholder

as the output, then this relationship is a function because there is only one maximum length (​

Placeholder

) associated with each given cost (​

Placeholder

).

Drag these tiles and drop them in the correct blank above

cost

maximum length

output

input

feet

Drag and Drop

What does it mean about the relationship between the x we used in function C and the x we use in this new function? This means that the ​

Placeholder

of function C is the ​

Placeholder

of the new function, and the ​

Placeholder

of function C is the input of the new function. The input and output have ​

Placeholder

places.

Drag these tiles and drop them in the correct blank above

input

output

switched

stayed

Drag and Drop

Suppose your guardians decided to increase their budget to $7,000. How would you use the value 7,000 with the inverse function, C^−1, to determine the maximum length of the home?

Because the ​

Placeholder

of C^−1 is the total cost and the ​

Placeholder

is the maximum length of the home, we should use 7,000 as the​

Placeholder

. The ​

Placeholder

is the maximum length of the home.

Drag these tiles and drop them in the correct blank above

input

output

​Equations for each material.
Check this with your answers.

Part 2: Reflecting on

Inverse Functions

Handout 2.4.B: Focus of f(x) = x³ + 1

Work on part A independently.

Swapping x and y in our tables is helpful in finding the inverse. What could
we do graphically to find the inverse?

Multiple Choice

What is the inverse of point (2, 9)?

• (9, 2)

• (-2, -9)

• (-9, -2)

Drag and Drop

What generalizations can we make about points that remain in the same location after we switch x and y? Any point where the x- and y-coordinates are the same, which is any point on the line y = x, will be a ​

Placeholder

​.

What generalizations can we make about points that remain in the same location after we switch x and y? All other points are ​

Placeholder

over the line y = x.

The graph of the inverse ​

Placeholder

a function because each ​

Placeholder

value is associated with only one ​

Placeholder

value. This demonstrates that it is the graph of a function.

Drag these tiles and drop them in the correct blank above

fixed point

reflected

is

input

output

is not

translated

dilated

Handout 2.4.C:
Reflecting on Your Own

Draw

Color this seahorse.

Part 3: Defining the

Inverse Function

Poll

Do all functions have inverse functions?

• Yes

• No

Reflections of each
Function

Are all these reflected equations
functions?

Multiple Choice

Are all these reflected equations
functions?

• Yes

• No

Multiple Choice

Which functions have graphs whose reflections over the line y = x are also functions? How do you know?

• Functions g and k. We can tell because each output is associated with only one input.

• Functions g and k. We can tell because each input is associated with only one output.

• Functions f and h. We can tell because each output is associated with only one input.

• Functions f and h. We can tell because each input is associated with only one output.

Multiple Choice

Functions g and k have reflections that are not functions. We can tell because each reflection has inputs that are associated with

• more than one output.

• only one output.

Multiple Choice

Is there a way to determine from the graph of a function whether the reflection over the line y = x will be a function?

• horizontal line test of original function

• vertical line test of original function

Restricted Domains 6.1.1

The graph of g(x) is shown.

If the domain of g(x) is restricted to x > 3,

graph its inverse function g⁻¹(x).

g(x)

Inverse Equation

Restricted g(x)

g⁻¹(x)

Restricted g(x)

and g⁻¹(x)

(Symmetric with

the line y = x)

Restricted Domains 6.1.2

Find the two possible inverse functions for the function,
f(x) = 4x² - 1 and f(x)’s respective restrictive domains.

y = 4x² - 1 let f(x) = y

x = 4y² - 1 swap x and y

x + 1 = 4y² add 1

¼(x+1) = y² divide 4

±√¼(x+1) = y square root

f⁻¹(x) = √¼(x+1)

f(x) Restricted Domain: (0, ∞)

f⁻¹(x) = -√¼(x+1)

f(x) Restricted Domain: (-∞, 0)

Key Takeaways

1.

All functions have inverse actions that undo them but not all
of these inverse actions are functions.

2.

A function that has output values associated with more than
one input value does not have an inverse function.

3.

A function and its inverse are opposites of each other.

4.

A function associates an input with an output: f(a) = b. If f has
an inverse function, then it associates the output with the
input: f ⁻¹(b) = a. This means that if you composed a function
and its inverse, you should get what you started with. That is,
f(f⁻¹(a)) = f(b)⁻¹ = a and f(f⁻¹(b)) = f (a) = b.

Formal Inverse Function Definition

Given a function f, its inverse f⁻¹(x) is defined by f(f⁻¹(x)) = f⁻¹(f(x)) for all values of x in the domain of f. That is, composing a function and its inverse, in any order, will yield the original input for every value in the domain of f.

Handout 2.4.D: Practice with Inverting

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin
the wheel as a class and spend a minute or
so discussing our answers.