Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Algebra 2
Nov 21, 2025
54
6 pages
Sam @sam.burchard
Function notation is a powerful mathematical language that helps you express relationships between inputs and outputs. Understanding how... Show more
A relation is a set of ordered pairs (x,y) that create graphs. The domain is the set of all x-values (inputs) of a relation, while the range is the set of all y-values (outputs). Think of domain as how "wide" the relation is and range as how "tall" it is.
A relation becomes a function when every element in the domain corresponds to exactly one value in the range. In other words, no x-value can be paired with multiple y-values. The vertical line test helps identify functions visually—if any vertical line intersects a graph more than once, it's not a function.
Some functions have an additional property called one-to-one (1-1), meaning every element in the range corresponds to exactly one element in the domain. Neither x nor y values repeat in a one-to-one function. The horizontal line test confirms this property—a function is one-to-one if no horizontal line crosses the graph more than once.
Remember Not all functions are one-to-one, but all one-to-one relations must first be functions. Master the vertical and horizontal line tests to quickly identify function properties on graphs!
Function notation uses expressions like f(x) (read as "f of x") to show that a relation is a function, where f(x) is equivalent to y. Different letters can be used for functions, like h(t) = t² - 5 or g(x) = x - 3, depending on the context.
When evaluating functions, you simply substitute the input value for the variable. For example, if f(x) = x² + 1, then f(3) = 3² + 1 = 10. This substitution process works with any input, including variables or expressions like f = ² + 1 = x² + 2x + 2.
Piecewise functions require careful attention to conditions. For instance, if f(x) equals 2x + 3 when x < -1 and equals 3 - x when x ≥ -1, you must check which condition your input satisfies before calculating. This explains why f(-5) = 2(-5) + 3 = -7, while f(-1) = 3 - (-1) = 4.
Pro Tip When working with composite functions or function operations like f(g(x)), always work from the inside out, substituting one step at a time to avoid errors.
Finite relations contain a limited number of ordered pairs, making them easier to analyze completely. When examining finite relations, we can list every ordered pair and precisely identify the domain and range.
To determine if a relation is a function, check whether each x-value (domain element) appears exactly once. For example, in the ordered pairs (1,4), (2,0), (3,5), each x-value (1, 2, and 3) appears exactly once, making this a function. Additionally, since each y-value (4, 0, and 5) appears exactly once, this is also a one-to-one function.
Not all functions are one-to-one, however. The relation with ordered pairs (1,6), (2,6), (3,6) is a function because each x-value appears once, but it's not one-to-one since the y-value 6 repeats. Relations like (1,4), (2,4), (2,6), (3,5) aren't functions at all because the x-value 2 is paired with multiple y-values.
Visual Check Arrow diagrams provide a quick way to verify functions—if any domain element has multiple arrows pointing from it, the relation is not a function!
Functions can model real-world scenarios like business profits. When working with related functions, you can solve for unknown functions using algebra. For instance, if profit p(x) equals revenue r(x) minus cost c(x), you can rearrange the equation to find c(x) = r(x) - p(x).
When combining functions, follow the rules of algebra carefully. For example, to find the product f(x)·g(x), multiply the functions together, and to find the sum f(x)+g(x), add them. These operations create new functions that may have different properties than the original functions.
Simplifying complex function expressions requires careful step-by-step work. For an expression like f(x)·g(x) - , substitute the function definitions, distribute terms, combine like terms, and express the result in standard form. This methodical approach ensures you don't miss terms or make sign errors.
Application Alert Functions model countless real-world scenarios—from profit calculations to physics equations. The ability to manipulate functions algebraically will serve you in many science and business applications!
Unlike finite relations, infinite relations contain unlimited ordered pairs, making them impossible to list completely. Instead, we describe their domains and ranges using set notation or interval notation like x ∈ ℝ (x is an element of all real numbers) or y ≥ -1.
The vertical line test still determines if an infinite relation is a function. If a relation passes this test, every x-value corresponds to exactly one y-value. Similarly, the horizontal line test checks if a function is one-to-one. For example, a linear function like y = x passes both tests and is therefore a one-to-one function.
Some common functions have specific domain and range patterns. For instance, square root functions like y = √ have restricted domains (x ≥ 2 in this case) and ranges (y ≥ 0). Absolute value functions like y = -|x+2| have full domains (x ∈ ℝ) but restricted ranges (y ≤ 0 here).
Notation Tip The symbols "∈" means "is an element of" and ℝ represents all real numbers. These symbols help express infinite domains and ranges concisely!
Linear functions like 2x - y = 6 have both domain and range extending across all real numbers. These functions create straight lines when graphed and pass both the vertical and horizontal line tests, making them one-to-one functions.
Absolute value functions like y = -|x + 2| create V-shaped graphs that are flipped upside down. They have domains of all real numbers, but their ranges are restricted (y ≤ 0 in this case). These functions pass the vertical line test but fail the horizontal line test, so they're functions but not one-to-one.
Square root functions like y = √ have restricted domains because you can't take the square root of negative numbers. For this function, the domain is x ≥ 2, and the range is y ≥ 0. Their curved shape passes the vertical and horizontal line tests, making them one-to-one functions.
Graphing Strategy Piecewise functions like f(x) = {x-2 if x<0, 1 if 0≤x≤2, 2x if x>2} require graphing each piece separately within its specified domain. Pay special attention to the transition points to ensure your graph is accurate!
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.
0
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Learn how to classify and solve systems of linear equations with two variables using graphing methods and common solutions.
Algebra 2Learn how to simplify radical expressions involving square roots, cube roots, and multiplication and division of radicals.
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
Sam
@sam.burchard
Function notation is a powerful mathematical language that helps you express relationships between inputs and outputs. Understanding how to work with functions is crucial for algebra, calculus, and many real-world applications you'll encounter in science and economics.
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A relation is a set of ordered pairs (x,y) that create graphs. The domain is the set of all x-values (inputs) of a relation, while the range is the set of all y-values (outputs). Think of domain as how "wide" the relation is and range as how "tall" it is.
A relation becomes a function when every element in the domain corresponds to exactly one value in the range. In other words, no x-value can be paired with multiple y-values. The vertical line test helps identify functions visually—if any vertical line intersects a graph more than once, it's not a function.
Some functions have an additional property called one-to-one (1-1), meaning every element in the range corresponds to exactly one element in the domain. Neither x nor y values repeat in a one-to-one function. The horizontal line test confirms this property—a function is one-to-one if no horizontal line crosses the graph more than once.
Remember: Not all functions are one-to-one, but all one-to-one relations must first be functions. Master the vertical and horizontal line tests to quickly identify function properties on graphs!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Function notation uses expressions like f(x) (read as "f of x") to show that a relation is a function, where f(x) is equivalent to y. Different letters can be used for functions, like h(t) = t² - 5 or g(x) = x - 3, depending on the context.
When evaluating functions, you simply substitute the input value for the variable. For example, if f(x) = x² + 1, then f(3) = 3² + 1 = 10. This substitution process works with any input, including variables or expressions like f = ² + 1 = x² + 2x + 2.
Piecewise functions require careful attention to conditions. For instance, if f(x) equals 2x + 3 when x < -1 and equals 3 - x when x ≥ -1, you must check which condition your input satisfies before calculating. This explains why f(-5) = 2(-5) + 3 = -7, while f(-1) = 3 - (-1) = 4.
Pro Tip: When working with composite functions or function operations like f(g(x)), always work from the inside out, substituting one step at a time to avoid errors.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Finite relations contain a limited number of ordered pairs, making them easier to analyze completely. When examining finite relations, we can list every ordered pair and precisely identify the domain and range.
To determine if a relation is a function, check whether each x-value (domain element) appears exactly once. For example, in the ordered pairs (1,4), (2,0), (3,5), each x-value (1, 2, and 3) appears exactly once, making this a function. Additionally, since each y-value (4, 0, and 5) appears exactly once, this is also a one-to-one function.
Not all functions are one-to-one, however. The relation with ordered pairs (1,6), (2,6), (3,6) is a function because each x-value appears once, but it's not one-to-one since the y-value 6 repeats. Relations like (1,4), (2,4), (2,6), (3,5) aren't functions at all because the x-value 2 is paired with multiple y-values.
Visual Check: Arrow diagrams provide a quick way to verify functions—if any domain element has multiple arrows pointing from it, the relation is not a function!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Functions can model real-world scenarios like business profits. When working with related functions, you can solve for unknown functions using algebra. For instance, if profit p(x) equals revenue r(x) minus cost c(x), you can rearrange the equation to find c(x) = r(x) - p(x).
When combining functions, follow the rules of algebra carefully. For example, to find the product f(x)·g(x), multiply the functions together, and to find the sum f(x)+g(x), add them. These operations create new functions that may have different properties than the original functions.
Simplifying complex function expressions requires careful step-by-step work. For an expression like f(x)·g(x) - , substitute the function definitions, distribute terms, combine like terms, and express the result in standard form. This methodical approach ensures you don't miss terms or make sign errors.
Application Alert: Functions model countless real-world scenarios—from profit calculations to physics equations. The ability to manipulate functions algebraically will serve you in many science and business applications!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Unlike finite relations, infinite relations contain unlimited ordered pairs, making them impossible to list completely. Instead, we describe their domains and ranges using set notation or interval notation like x ∈ ℝ (x is an element of all real numbers) or y ≥ -1.
The vertical line test still determines if an infinite relation is a function. If a relation passes this test, every x-value corresponds to exactly one y-value. Similarly, the horizontal line test checks if a function is one-to-one. For example, a linear function like y = x passes both tests and is therefore a one-to-one function.
Some common functions have specific domain and range patterns. For instance, square root functions like y = √ have restricted domains (x ≥ 2 in this case) and ranges (y ≥ 0). Absolute value functions like y = -|x+2| have full domains (x ∈ ℝ) but restricted ranges (y ≤ 0 here).
Notation Tip: The symbols "∈" means "is an element of" and ℝ represents all real numbers. These symbols help express infinite domains and ranges concisely!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Linear functions like 2x - y = 6 have both domain and range extending across all real numbers. These functions create straight lines when graphed and pass both the vertical and horizontal line tests, making them one-to-one functions.
Absolute value functions like y = -|x + 2| create V-shaped graphs that are flipped upside down. They have domains of all real numbers, but their ranges are restricted (y ≤ 0 in this case). These functions pass the vertical line test but fail the horizontal line test, so they're functions but not one-to-one.
Square root functions like y = √ have restricted domains because you can't take the square root of negative numbers. For this function, the domain is x ≥ 2, and the range is y ≥ 0. Their curved shape passes the vertical and horizontal line tests, making them one-to-one functions.
Graphing Strategy: Piecewise functions like f(x) = {x-2 if x<0, 1 if 0≤x≤2, 2x if x>2} require graphing each piece separately within its specified domain. Pay special attention to the transition points to ensure your graph is accurate!
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.
0
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Learn how to classify and solve systems of linear equations with two variables using graphing methods and common solutions.
Algebra 2Learn how to simplify radical expressions involving square roots, cube roots, and multiplication and division of radicals.
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
