Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Algebra 2
Dec 2, 2025
121
4 pages
Trevor @trevk
Functions are a fundamental concept in mathematics that describe relationships where each input produces exactly one output. They're... Show more
Functions are special types of relationships where each input value corresponds to exactly one output value. Think of a function as a machine you put in a value (input), and get exactly one value back (output).
A relation is simply a set of ordered pairs (x,y), where x is the input and y is the output. Not all relations are functions though! A relation becomes a function only when each x-value is paired with exactly one y-value. If an x-value has multiple y-values, the relation is not a function.
When working with functions, we often use function notation written as f(x), which is read as "f of x." This notation helps us identify the dependent variable (output) in terms of the independent variable (input).
Quick Check To determine if a relation is a function, ask yourself "Does each x-value have exactly one y-value?" If you see any x-value with multiple outputs, it's not a function!
The domain of a function is the set of all possible input values , while the range is the set of all possible output values . These help us understand what values can go into and come out of our function.
Functions let you calculate output values when you know the inputs. To evaluate a function, simply substitute the given input value for x in the function rule.
For example, with f(x) = 3x - 4, finding f(2) means replacing x with 2 f(2) = 3(2) - 4 = 6 - 4 = 2. This process works for any function, whether it's linear, quadratic, or more complex.
Real-world relationships can be analyzed to determine if they're functions. A key question to ask "Can each input have multiple outputs?" If yes, it's not a function. For example, a person can have multiple phone numbers, so the relationship between people and phone numbers isn't a function.
Remember When deciding if a real-world situation represents a function, think about whether each input value (like a person) can correspond to more than one output value (like multiple phone numbers).
Functions are powerful tools for modeling practical situations. They help us understand how changes in one variable affect another, making them essential for describing everything from phone bills to population growth.
Graphs give us a visual way to understand functions. The Vertical Line Test is your best friend when determining if a graph represents a function if any vertical line intersects the graph more than once, it's not a function.
When describing domains and ranges, we use set notation. For example, {x|x ∈ ℝ} means "the set of all x-values such that x is a real number" - basically saying x can be any real number. The domain tells us what x-values we can use, while the range shows what y-values we can get.
Different function types have characteristic shapes and domains/ranges
Pro Tip Your graphing calculator is a powerful tool for visualizing functions and understanding their domains and ranges. Always use it to check your work when graphing complex functions!
Recognizing function types by their graphs helps you quickly determine their key characteristics without having to analyze every point individually. With practice, you'll identify patterns at a glance.
Graphing calculators are amazing tools for evaluating functions. Just enter the function, input your x-value, and the calculator gives you the corresponding y-value. This saves time and reduces calculation errors when working with complex expressions.
Real-world situations can often be modeled with functions. For example, a gift shop selling specialty mix at 4.95 for the tin can be represented as c(x) = 2.99x + 4.95, where x is the number of pounds. This lets you calculate costs for any amount.
Linear functions are especially useful for modeling everyday situations like
Make It Real Try creating function models for everyday situations, like the cost of streaming services or the distance a car travels over time. This practice helps math concepts stick!
When solving real-world problems, identifying what the variables represent is crucial. Always label your variables clearly and include proper units in your answers. This ensures your solutions make practical sense in the real world.
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.
1
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Learn how to complete a table, graph solution sets, and state domain and range using interval notation in inequality notation.
Algebra 2Notes about the topic
Algebra 2This is about adding, subtracting, multiplying and dividing complex numbers.
Algebra 2Conjugating Root Theorem and Fundamental Theorem of Algebra
Algebra 2Complex numbers, solving quadratics, perfect square trinomials
Algebra 2Notes and tips to apply Quadratic formula.
Algebra 2App 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
Trevor
@trevk
Functions are a fundamental concept in mathematics that describe relationships where each input produces exactly one output. They're essential for modeling real-world situations and form the foundation of higher math concepts you'll encounter 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
Functions are special types of relationships where each input value corresponds to exactly one output value. Think of a function as a machine: you put in a value (input), and get exactly one value back (output).
A relation is simply a set of ordered pairs (x,y), where x is the input and y is the output. Not all relations are functions though! A relation becomes a function only when each x-value is paired with exactly one y-value. If an x-value has multiple y-values, the relation is not a function.
When working with functions, we often use function notation written as f(x), which is read as "f of x." This notation helps us identify the dependent variable (output) in terms of the independent variable (input).
Quick Check: To determine if a relation is a function, ask yourself: "Does each x-value have exactly one y-value?" If you see any x-value with multiple outputs, it's not a function!
The domain of a function is the set of all possible input values , while the range is the set of all possible output values . These help us understand what values can go into and come out of our function.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Functions let you calculate output values when you know the inputs. To evaluate a function, simply substitute the given input value for x in the function rule.
For example, with f(x) = 3x - 4, finding f(2) means replacing x with 2: f(2) = 3(2) - 4 = 6 - 4 = 2. This process works for any function, whether it's linear, quadratic, or more complex.
Real-world relationships can be analyzed to determine if they're functions. A key question to ask: "Can each input have multiple outputs?" If yes, it's not a function. For example, a person can have multiple phone numbers, so the relationship between people and phone numbers isn't a function.
Remember: When deciding if a real-world situation represents a function, think about whether each input value (like a person) can correspond to more than one output value (like multiple phone numbers).
Functions are powerful tools for modeling practical situations. They help us understand how changes in one variable affect another, making them essential for describing everything from phone bills to population growth.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Graphs give us a visual way to understand functions. The Vertical Line Test is your best friend when determining if a graph represents a function: if any vertical line intersects the graph more than once, it's not a function.
When describing domains and ranges, we use set notation. For example, {x|x ∈ ℝ} means "the set of all x-values such that x is a real number" - basically saying x can be any real number. The domain tells us what x-values we can use, while the range shows what y-values we can get.
Different function types have characteristic shapes and domains/ranges:
Pro Tip: Your graphing calculator is a powerful tool for visualizing functions and understanding their domains and ranges. Always use it to check your work when graphing complex functions!
Recognizing function types by their graphs helps you quickly determine their key characteristics without having to analyze every point individually. With practice, you'll identify patterns at a glance.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Graphing calculators are amazing tools for evaluating functions. Just enter the function, input your x-value, and the calculator gives you the corresponding y-value. This saves time and reduces calculation errors when working with complex expressions.
Real-world situations can often be modeled with functions. For example, a gift shop selling specialty mix at 4.95 for the tin can be represented as c(x) = 2.99x + 4.95, where x is the number of pounds. This lets you calculate costs for any amount.
Linear functions are especially useful for modeling everyday situations like:
Make It Real: Try creating function models for everyday situations, like the cost of streaming services or the distance a car travels over time. This practice helps math concepts stick!
When solving real-world problems, identifying what the variables represent is crucial. Always label your variables clearly and include proper units in your answers. This ensures your solutions make practical sense in the real world.
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.
1
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Learn how to complete a table, graph solution sets, and state domain and range using interval notation in inequality notation.
Algebra 2Notes about the topic
Algebra 2This is about adding, subtracting, multiplying and dividing complex numbers.
Algebra 2Conjugating Root Theorem and Fundamental Theorem of Algebra
Algebra 2Complex numbers, solving quadratics, perfect square trinomials
Algebra 2Notes and tips to apply Quadratic formula.
Algebra 2App 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
Algebra 1
Algebra
Calculus 1
Overige