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Understanding Functions: Easy Step-by-Step Guide to Inverses, Compositions, and Transformations

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Mira

6/12/2023

Algebra 2

functions & relations, including transformations

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Algebra 2

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Transformations

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Putting It All Together

Understanding Functions: Easy Step-by-Step Guide to Inverses, Compositions, and Transformations

A comprehensive guide to understanding inverses and compositions of functions, covering essential mathematical concepts and transformations.

• Function transformations include translations (vertical and horizontal shifts), reflections across axes, and vertical dilations (stretches and compressions)
• Relations and functions are distinguished by their input-output relationships, with functions having exactly one output for each input
• Composition of functions involves using one function's output as another's input, while inverse functions swap domain and range values
• The guide emphasizes practical applications through examples and verification methods
• Understanding domain and range restrictions is crucial for both function compositions and inverses

...

6/12/2023

284

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Subjects

Algebra 2

Transformations

Putting It All Together

 

Algebra 2

•

284

•

Jun 12, 2023

•

1 page

Understanding Functions: Easy Step-by-Step Guide to Inverses, Compositions, and Transformations

user profile picture

Mira

@miraa._noui

A comprehensive guide to understanding inverses and compositions of functions, covering essential mathematical concepts and transformations.

• Function transformations include translations (vertical and horizontal shifts), reflections across axes, and vertical dilations (stretches and compressions)
• Relations and functions are... Show more

final Study GUIDE transformations 1. translations: + vertical: f(x) = k ; horizontal: f(x ±h) 2. reflections. →-f(x): x-axis, opposite y-val

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Function Transformations and Relations

This comprehensive page covers fundamental concepts of function transformations, relations, and function compositions. The content explores various mathematical operations and their applications in graphing and problem-solving.

Definition: A function is a special type of relation where each input has exactly one output, while a relation can have multiple outputs for a single input.

Highlight: The difference between relations and functions in math is crucial for understanding mathematical relationships.

Example: When finding inverse functions, swap x and y coordinates and solve for y. For instance, given fxx = -3x + 6, replace x with y and y with x, then solve for y to find the inverse.

Vocabulary:

  • Domain: The set of all possible input xx values
  • Range: The set of all possible output yy values
  • Interval notation: A mathematical way to represent sets using brackets []

Definition: The step-by-step guide to function transformations includes:

  1. Translations verticalshiftsusingf(xvertical shifts using f(x ± k, horizontal shifts using fx±hx ± h)
  2. Reflections −f(x-f(x for x-axis reflection, f−x-x for y-axis reflection)
  3. Vertical dilations stretcheswithfactor>1,compressionswith0<factor<1stretches with factor > 1, compressions with 0 < factor < 1

Highlight: For composition of functions, ensure the domain of the inner function matches the required input of the outer function.

Example: To verify if functions are inverses, check if fg(xg(x) = x and gf(xf(x) = x. For instance, verifying 3x-1 and 3x+3 are inverses requires substituting one into the other.

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I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

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Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

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Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

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The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

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In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

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I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

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Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

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THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

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This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user