Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Math
Dec 15, 2025
1,090
13 pages
Emerald Shane Timbas @emeraldshanetim
Functions are like mathematical machines that take inputs and give you specific outputs. Understanding how they work and... Show more
Think of a relation as a collection of connections between inputs and outputs, written as ordered pairs (x, y). The domain contains all your possible inputs , while the range holds all possible outputs .
A function is a special type of relation where each input gives you exactly one output. It's like having a vending machine - you press one button (input) and get exactly one specific item (output).
Key Point Functions follow the "one input, one output" rule - no exceptions!
Let's see this in action! In the first example, each letter (L, O, V, E) connects to exactly one number - that's a function. It follows our golden rule perfectly.
But look at Gerald's situation - he's connected to six different people! Since one input (Gerald) has multiple outputs, this is not a function. Functions don't work that way.
Remember If any input has more than one output, you don't have a function!
You can represent functions in three main ways. Mapping diagrams use arrows to show connections between inputs and outputs. Tables of values organize your x and y pairs in neat rows and columns.
Graphs give you the visual picture, and here's where the vertical line test becomes your best friend. Draw any vertical line on the graph - if it crosses the function more than once, you don't have a function.
Each method has its strengths, so pick the one that makes the most sense for your specific problem.
Pro Tip The vertical line test is the fastest way to check if a graph represents a function!
Relations come in four flavors. One-to-one means each input connects to exactly one output, and each output connects to exactly one input. Many-to-one has multiple inputs sharing the same output.
One-to-many connects one input to several outputs (this breaks the function rule). Many-to-many is the most complex, with multiple inputs connecting to multiple outputs.
Only one-to-one and many-to-one relations qualify as functions. The other two violate our fundamental function rule.
Quick Check Functions can only be one-to-one or many-to-one!
Function evaluation means substituting a specific value for x and calculating the result. When you see f(2), you're plugging in 2 for every x in your function.
Take f(x) = 3x + 1. To find f(2), replace x with 2 f(2) = 3(2) + 1 = 7. It's like following a recipe - substitute the ingredient and follow the steps.
Practice makes perfect here. The more you substitute and calculate, the faster you'll become at evaluating any function.
Success Strategy Always substitute first, then follow the order of operations carefully!
Sometimes you'll evaluate functions with expressions instead of simple numbers. For f(x) = x² + 2 and x = 2 - x, you substitute the entire expression for every x.
Use the FOIL method when expanding expressions like ². First, Outside, Inside, Last - this systematic approach prevents mistakes.
Don't rush through these problems. Take your time with each step, and double-check your algebra as you go.
FOIL Reminder First terms, Outside terms, Inside terms, Last terms - then combine like terms!
Function addition combines two functions by adding their expressions together. Write (x) = f(x) + g(x), then substitute both function expressions.
The key is organizing your work clearly. Write out both functions, add them together, then combine like terms systematically. Group x³ terms, x² terms, x terms, and constants separately.
Start with simpler examples before tackling complex polynomials. This builds your confidence and helps you spot patterns.
Organization Tip Line up like terms vertically to make combining easier!
Function subtraction follows similar steps, but you need to distribute the negative sign carefully. When you see (x), you're really doing f(x) + .
Change the sign of every term in the second function, then combine like terms. This is where many students make mistakes, so double-check that you've flipped all the signs correctly.
Practice with the same functions using both addition and subtraction - you'll see how the operations relate to each other.
Sign Alert Distribute that negative sign to every term in the second function!
Here's another subtraction example to solidify your understanding. Notice how - becomes + after distributing the negative.
The final answer -2x² - 4x + 17 comes from carefully combining 5 + 12 = 17, -x² + = -2x², and -4x stays as is.
Always verify your work by checking that you've handled every term correctly. One small sign error can throw off your entire answer.
Double-Check Count the terms in your original expressions and make sure they all appear in your final answer!
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
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Google Play
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
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
Emerald Shane Timbas
@emeraldshanetim
Functions are like mathematical machines that take inputs and give you specific outputs. Understanding how they work and how to manipulate them is crucial for succeeding in algebra and beyond.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Think of a relation as a collection of connections between inputs and outputs, written as ordered pairs (x, y). The domain contains all your possible inputs , while the range holds all possible outputs .
A function is a special type of relation where each input gives you exactly one output. It's like having a vending machine - you press one button (input) and get exactly one specific item (output).
Key Point: Functions follow the "one input, one output" rule - no exceptions!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Let's see this in action! In the first example, each letter (L, O, V, E) connects to exactly one number - that's a function. It follows our golden rule perfectly.
But look at Gerald's situation - he's connected to six different people! Since one input (Gerald) has multiple outputs, this is not a function. Functions don't work that way.
Remember: If any input has more than one output, you don't have a function!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
You can represent functions in three main ways. Mapping diagrams use arrows to show connections between inputs and outputs. Tables of values organize your x and y pairs in neat rows and columns.
Graphs give you the visual picture, and here's where the vertical line test becomes your best friend. Draw any vertical line on the graph - if it crosses the function more than once, you don't have a function.
Each method has its strengths, so pick the one that makes the most sense for your specific problem.
Pro Tip: The vertical line test is the fastest way to check if a graph represents a function!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Relations come in four flavors. One-to-one means each input connects to exactly one output, and each output connects to exactly one input. Many-to-one has multiple inputs sharing the same output.
One-to-many connects one input to several outputs (this breaks the function rule). Many-to-many is the most complex, with multiple inputs connecting to multiple outputs.
Only one-to-one and many-to-one relations qualify as functions. The other two violate our fundamental function rule.
Quick Check: Functions can only be one-to-one or many-to-one!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Function evaluation means substituting a specific value for x and calculating the result. When you see f(2), you're plugging in 2 for every x in your function.
Take f(x) = 3x + 1. To find f(2), replace x with 2: f(2) = 3(2) + 1 = 7. It's like following a recipe - substitute the ingredient and follow the steps.
Practice makes perfect here. The more you substitute and calculate, the faster you'll become at evaluating any function.
Success Strategy: Always substitute first, then follow the order of operations carefully!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sometimes you'll evaluate functions with expressions instead of simple numbers. For f(x) = x² + 2 and x = 2 - x, you substitute the entire expression for every x.
Use the FOIL method when expanding expressions like ². First, Outside, Inside, Last - this systematic approach prevents mistakes.
Don't rush through these problems. Take your time with each step, and double-check your algebra as you go.
FOIL Reminder: First terms, Outside terms, Inside terms, Last terms - then combine like terms!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Function addition combines two functions by adding their expressions together. Write (x) = f(x) + g(x), then substitute both function expressions.
The key is organizing your work clearly. Write out both functions, add them together, then combine like terms systematically. Group x³ terms, x² terms, x terms, and constants separately.
Start with simpler examples before tackling complex polynomials. This builds your confidence and helps you spot patterns.
Organization Tip: Line up like terms vertically to make combining easier!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Function subtraction follows similar steps, but you need to distribute the negative sign carefully. When you see (x), you're really doing f(x) + .
Change the sign of every term in the second function, then combine like terms. This is where many students make mistakes, so double-check that you've flipped all the signs correctly.
Practice with the same functions using both addition and subtraction - you'll see how the operations relate to each other.
Sign Alert: Distribute that negative sign to every term in the second function!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Here's another subtraction example to solidify your understanding. Notice how - becomes + after distributing the negative.
The final answer -2x² - 4x + 17 comes from carefully combining: 5 + 12 = 17, -x² + = -2x², and -4x stays as is.
Always verify your work by checking that you've handled every term correctly. One small sign error can throw off your entire answer.
Double-Check: Count the terms in your original expressions and make sure they all appear in your final answer!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
22
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
Google Play
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
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
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