Posted on

## Finding an Inverse Function

Functions take x values to y values. Inverse functions reverse the direction. Now y values go to x values.

Some equations are easy to find the inverse of others because it is easy to see how to reverse the function for all values.

Here are 3 examples of increasing complexity:

1. If f(x) = x + 3, then we can reverse the function by solving x = f(x) + 3 for f(x), which is f(x) = x – 3.

2. If f(x) = 5x, then we can reverse the function by solving x = 5 f(x) for f(x), which is $$f(x) = \frac{x}{5}$$

3. If f(x)=x3, then we can reverse the function by solving for x = (f(x))3 which is

$$f(x) = \sqrt[3]{x}$$

### Same Content from OpenStax

We can now consider one-to-one functions and show how to find their inverses. Recall that a function maps elements in the domain of f to elements in the range of f. The inverse function maps each element from the range of f back to its corresponding element from the domain of f. Therefore, to find the inverse function of a one-to-one function f, given any y in the range of f, we need to determine which x in the domain of f satisfies f(x)=y. Since f is one-to-one, there is exactly one such value x. We can find that value x by solving the equation f(x)=y for x. Doing so, we are able to write x as a function of y where the domain of this function is the range of f and the range of this new function is the domain of f. Consequently, this function is the inverse of f, and we write x=f−1(y). Since we typically use the variable x to denote the independent variable and y to denote the dependent variable, we often interchange the roles of x and y, and write y=f−1(x). Representing the inverse function in this way is also helpful later when we graph a function f and its inverse f−1 on the same axes.

PROBLEM-SOLVING STRATEGY: FINDING AN INVERSE FUNCTION

1. Solve the equation y=f(x) for x.
2. Interchange the variables x and y and write y=f−1(x).
Posted on

## STEAM Exploration: Bridge & Tower

Towers and bridges both need strong shapes. Start with this video to learn about strong shapes. This video will help you imagine great designs when you draw plans for the challenges below.

This video shows why triangles are a good shape to use when building a tall tower.

We have many tower-building challenges. We even have a page focused on our experiences building towers. Check out edMe’s Tower Page any time!

## Materials

We sell tower kits and bridge kits, and you can get these materials around your house too! Here is a quick overview of the best options:

• Spaghetti and Marshmallows – Not only delicious, but easy for the youngest architects. Can be sticky so a sheet or big piece of paper is a worthwhile work area. Some people use rubber bands so it’s easier to use these materials over and over again.
• Straws and tape – Easy to build although it takes some team work to redesign (scissors involved). Definitely allows for gigantic designs.
• Legos – If you have them, give it a shot! It takes a lot of legos to build a BIG tower or bridge, but the possibilities are definitely there.
• Newspaper – Adds a unique spin to the challenges because architects must first figure out what shape of newspaper is the strongest. The ink easily washes off of hands and it allows for some of the largest builds.
• Cardboard – If the delivery truck stops often at your house, consider injecting STEM into your recyclables. We find it’s most engaging to flatten or break down the boxes first, so you have sheets of cardboard that allow the imagination to explore. (May need a good pair of scissors.)

## Freemium Virtual Simulator Apps

We have a series of challenges for you to engage with towers and bridges this week. Many of them use physical materials that you can buy here or at a local store. And, you will want to download the free app:

## Challenges

• Challenge 13: Build the tallest tower possible with 20 straws and tape.
• Challenge 14: Build the first 5 bridges in the app. (Click here if you have trouble installing it!)
• Challenge 15: Build a bridge tower in real life.
• Challenge 16: Building a toothpick tower that can support a brick.
• Challenge 17: Becoming an accomplished virtual bridge builder. (Challenges 6-10 in the app)
• Challenge 18: Understand and celebrate a bridge in your area.
• Challenge 19: Building a tall bridge
• Challenge 20: Investigate a historic bridge and show why it is strong.
• Challenge 21: Building a long bridge
• Challenge 22: Becoming an advanced virtual bridge builder. (Challenges 11-15 in the app)
• Challenge 23: Becoming an expert virtual bridge builder! (Challenges 16-20 in the app)
• Challenge 24: Create a video that teaches someone younger why bridges are strong
Posted on

## Create your Harry Potter and the Sorcerer’s Stone cover

Covers are great because they summarize the theme of the book. What was the theme of this story in your opinion?

One thing we all hate are spoilers! Spoilers give away exciting details in the story. There were many events that were fun to read about in this book! If someone told you what was going to happen before you read, you would not have enjoyed reading as much.

When you create your cover, avoid spoilers!

Challenge:

1. Write your theme on a piece of paper.
2. Write an event that you do not want to spoil for the reader.
3. Draw a cover that expresses that theme.

We will post feedback to your account soon afterwards!

Posted on

## Cover Rubric for Harry Potter and the Sorcerer’s Stone

Remember, themes cover the entire book. Your cover should show the theme in pictures. Also, covers are exciting, so think about something you found exciting in the book. Would you be excited to read the book if you saw this cover?

Use this rubric to help you think about your cover. If you want feedback, you can submit how you think your cover addresses these criteria and we will give you feedback. Great work on this book!

 Cover is missing or could be the cover of any book. Cover does not show themes, main characters, or important events this book. Cover shows a theme, but is not clear enough for someone else to understand a theme in this book. Cover is clear, exciting, and shows a theme within this book. Missing 0 1 2

Review your work and give yourself a score. Explain why you gave your work this score.

Posted on

## Event Rubric for Harry Potter and the Sorcerer’s Stone

Themes cover the entire book. It’s the message. Authors develop theme through their characters, the characters’ actions, what happens as a result, and even word choice and writing style help make a theme. What was the theme for this book? This rubric shows how you can evaluate your work on describing the theme.

 Attempt is missing or barely started. Event is incorrect or not clearly described. Event is somewhat unclear or may not be a true spoiler (If your event is in the beginning of the book or not that interesting, then it may not worry people if they know it before they start reading.) Event is clearly described and is certainly something a reader wants to discover on their own! Missing 0 1 2

Review your work and give yourself a score. Explain why you gave your work this score.

Posted on

## Rubric for Theme to Harry Potter and the Sorcerer’s Stone

Themes cover the entire book. It’s the message. Authors develop theme through their characters, the characters’ actions, what happens as a result, and even word choice and writing style help make a theme. What was the theme for this book? This rubric shows how you can evaluate your work on describing the theme.

 Theme is missing or barely started. The theme is unclear for another reader, or may misinterpret the message within this book. The theme may cover one or two events, but not the whole book. Or, it could need more description so someone else can understand it. The theme addresses events and characters from the entire book. Bonus points if you cite the events that support your theme! Missing 0 1 2

Review your work and give yourself a score. Explain why you gave your work this score.

Posted on

## Catapults

Use your kit to make things fly! We give you a penny to launch, and you can find lots of other things to launch in your catapult.

Directions:

1. Create a strong base by tying together some sticks.