Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Math
Dec 13, 2025
695
6 pages
sam โ๐โหโนโก @samantha.badiola_
Ever wonder how math connects to real life? Functions and relations are everywhere - from calculating your phone... Show more
Think of a relation as a matchmaker that connects values from one group (the domain) to another group (the range). It's basically a set of ordered pairs like (x,y) where x gets paired with y.
Now here's where it gets interesting - a function is a special type of relation with one important rule each x-value can only be matched with one y-value. No cheating allowed! If you see an x-value trying to connect to multiple y-values, it's not a function.
For example, f = {(1,2), (2,3), (3,5)} is a function because each x has its own unique y. But g = {(1,3), (1,4), (2,5)} isn't a function because x = 1 is trying to pair with both 3 and 4.
Quick Tip Functions use the notation f(x) = y, which reads as "f of x equals y." This just means when you input x, you get y as the output.
Mapping diagrams show functions as arrows connecting domain values to range values. If any domain value has multiple arrows pointing out, it's not a function - remember, one input can only have one output!
The vertical line test is your best friend for checking graphs. Draw imaginary vertical lines across the graph - if any line touches the curve more than once, you don't have a function. It's that simple!
Tables of values organize functions neatly, showing x and y pairs in columns. These make it super easy to see the relationship between inputs and outputs at a glance.
Piecewise functions are like having different rules for different situations. They use multiple equations depending on which domain you're working in - think of them as "if-then" statements in math form.
Remember Functions are everywhere in real life - from calculating grades based on test scores to determining taxi fares based on distance traveled!
Evaluating functions means finding the output when you plug in a specific input value. It's like following a recipe - substitute the given number for x and calculate the result.
For regular functions like f(x) = xยฒ - 4x + 4, just replace every x with your given number. If you need f(2), substitute 2 for x f(2) = 2ยฒ - 4(2) + 4 = 0. Easy math!
Piecewise functions require an extra step - first figure out which equation to use based on your x-value, then substitute and solve. Check the conditions to pick the right formula.
Practice makes perfect with these problems. Start with simple substitutions, then work your way up to more complex piecewise functions as you build confidence.
Pro Strategy Always double-check which piece of a piecewise function applies to your x-value before calculating - this prevents the most common mistakes!
Just like regular numbers, functions can be added, subtracted, multiplied, and divided. The key is performing the operation on the function expressions themselves.
For addition and subtraction, combine like terms (x) = f(x) + g(x). If f(x) = x+1 and g(x) = xยฒ+2x-8, then (x) = + = xยฒ + 3x - 7.
Multiplication works the same way (fรg)(x) = f(x) ร g(x). Just multiply the entire expressions together and simplify. Division follows suit, but watch out for values that make the denominator zero!
Composite functions like (fโg)(x) = f(g(x)) mean you're plugging one function into another. Work from the inside out - first find g(x), then use that result as the input for f(x).
Memory Hack Think of composite functions like nesting dolls - you work your way from the innermost function outward!
Composite functions might look intimidating, but they're just functions within functions. The notation (gโf)(x) means "g of f of x" - you calculate f(x) first, then plug that answer into g(x).
Let's say f(x) = x + 3 and g(x) = 2x - 7. To find (gโf)(x), substitute the entire f(x) expression into g(x) g(f(x)) = g = 2 - 7 = 2x + 6 - 7 = 2x - 1.
Notice that (gโf)(x) usually gives a different result than (fโg)(x) - order matters! Always work carefully through each step and double-check your algebra.
These composite functions show up everywhere in real applications - like converting temperature from Celsius to Fahrenheit, then adjusting for altitude effects.
Success Tip When dealing with composite functions, take your time with substitution and use parentheses to avoid sign errors!
Functions aren't just abstract math - they're powerful tools for modeling real-world relationships! Mathematical modeling helps us understand how different variables affect each other in everyday situations.
Direct variation shows relationships where both variables increase or decrease together - like distance and time when driving at constant speed. Inverse variation describes opposite relationships - as one increases, the other decreases, like speed and travel time.
Joint variation involves three variables where z depends on both x and y together - think about how the area of a rectangle depends on both length and width.
The constant of variation (k) is crucial - it tells you exactly how strong the relationship is. Once you find k using given values, you can predict what happens in new situations using your mathematical model.
Real-Life Connection These variations explain everything from how your electricity bill relates to usage, to how gravitational force changes with distance between objects!
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 โ all at your fingertips.
22
Smart Tools NEW
Transform this note into: โ 50+ Practice Questions โ Interactive Flashcards โ Full Mock Exam โ Essay Outlines
Master the three forms of quadratic equations and their key properties
MathApp 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 itโs very much worth it and youโll 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 donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve 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 help๐ค๐ค
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.๐๐ฉท๐
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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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 Iโve 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 itโs very much worth it and youโll 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 donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve 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 help๐ค๐ค
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.๐๐ฉท๐
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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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 Iโve seen! Geometry too!
Paul T
iOS user
sam โ๐โหโนโก
@samantha.badiola_
Ever wonder how math connects to real life? Functions and relations are everywhere - from calculating your phone bill to predicting population growth. These mathematical tools help us understand patterns and solve everyday problems in a systematic way.
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 matchmaker that connects values from one group (the domain) to another group (the range). It's basically a set of ordered pairs like (x,y) where x gets paired with y.
Now here's where it gets interesting - a function is a special type of relation with one important rule: each x-value can only be matched with one y-value. No cheating allowed! If you see an x-value trying to connect to multiple y-values, it's not a function.
For example, f = {(1,2), (2,3), (3,5)} is a function because each x has its own unique y. But g = {(1,3), (1,4), (2,5)} isn't a function because x = 1 is trying to pair with both 3 and 4.
Quick Tip: Functions use the notation f(x) = y, which reads as "f of x equals y." This just means when you input x, you get y as the output.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Mapping diagrams show functions as arrows connecting domain values to range values. If any domain value has multiple arrows pointing out, it's not a function - remember, one input can only have one output!
The vertical line test is your best friend for checking graphs. Draw imaginary vertical lines across the graph - if any line touches the curve more than once, you don't have a function. It's that simple!
Tables of values organize functions neatly, showing x and y pairs in columns. These make it super easy to see the relationship between inputs and outputs at a glance.
Piecewise functions are like having different rules for different situations. They use multiple equations depending on which domain you're working in - think of them as "if-then" statements in math form.
Remember: Functions are everywhere in real life - from calculating grades based on test scores to determining taxi fares based on distance traveled!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Evaluating functions means finding the output when you plug in a specific input value. It's like following a recipe - substitute the given number for x and calculate the result.
For regular functions like f(x) = xยฒ - 4x + 4, just replace every x with your given number. If you need f(2), substitute 2 for x: f(2) = 2ยฒ - 4(2) + 4 = 0. Easy math!
Piecewise functions require an extra step - first figure out which equation to use based on your x-value, then substitute and solve. Check the conditions to pick the right formula.
Practice makes perfect with these problems. Start with simple substitutions, then work your way up to more complex piecewise functions as you build confidence.
Pro Strategy: Always double-check which piece of a piecewise function applies to your x-value before calculating - this prevents the most common mistakes!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Just like regular numbers, functions can be added, subtracted, multiplied, and divided. The key is performing the operation on the function expressions themselves.
For addition and subtraction, combine like terms: (x) = f(x) + g(x). If f(x) = x+1 and g(x) = xยฒ+2x-8, then (x) = + = xยฒ + 3x - 7.
Multiplication works the same way: (fรg)(x) = f(x) ร g(x). Just multiply the entire expressions together and simplify. Division follows suit, but watch out for values that make the denominator zero!
Composite functions like (fโg)(x) = f(g(x)) mean you're plugging one function into another. Work from the inside out - first find g(x), then use that result as the input for f(x).
Memory Hack: Think of composite functions like nesting dolls - you work your way from the innermost function outward!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Composite functions might look intimidating, but they're just functions within functions. The notation (gโf)(x) means "g of f of x" - you calculate f(x) first, then plug that answer into g(x).
Let's say f(x) = x + 3 and g(x) = 2x - 7. To find (gโf)(x), substitute the entire f(x) expression into g(x): g(f(x)) = g = 2 - 7 = 2x + 6 - 7 = 2x - 1.
Notice that (gโf)(x) usually gives a different result than (fโg)(x) - order matters! Always work carefully through each step and double-check your algebra.
These composite functions show up everywhere in real applications - like converting temperature from Celsius to Fahrenheit, then adjusting for altitude effects.
Success Tip: When dealing with composite functions, take your time with substitution and use parentheses to avoid sign errors!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Functions aren't just abstract math - they're powerful tools for modeling real-world relationships! Mathematical modeling helps us understand how different variables affect each other in everyday situations.
Direct variation shows relationships where both variables increase or decrease together - like distance and time when driving at constant speed. Inverse variation describes opposite relationships - as one increases, the other decreases, like speed and travel time.
Joint variation involves three variables where z depends on both x and y together - think about how the area of a rectangle depends on both length and width.
The constant of variation (k) is crucial - it tells you exactly how strong the relationship is. Once you find k using given values, you can predict what happens in new situations using your mathematical model.
Real-Life Connection: These variations explain everything from how your electricity bill relates to usage, to how gravitational force changes with distance between objects!
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 โ all at your fingertips.
22
Smart Tools NEW
Transform this note into: โ 50+ Practice Questions โ Interactive Flashcards โ Full Mock Exam โ Essay Outlines
Master the three forms of quadratic equations and their key properties
MathApp 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 itโs very much worth it and youโll 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 donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve 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 help๐ค๐ค
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.๐๐ฉท๐
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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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 Iโve 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 itโs very much worth it and youโll 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 donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve 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 help๐ค๐ค
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.๐๐ฉท๐
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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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 Iโve seen! Geometry too!
Paul T
iOS user
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