Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
GenMath
Dec 13, 2025
1,123
5 pages
bossing @coteeee
Welcome to your guide on Rational, Exponential, and Logarithmic Functions! These powerful mathematical concepts appear in everything from... Show more
When tackling mathematics problems, being able to identify different function types is essential. Rational functions involve fractions with variables, such as . They often appear when calculating how resources relate to tasks.
Exponential functions have variables in the exponent position and model rapid growth scenarios like bacteria multiplication. These functions always have a positive range and appear in real-world applications like population growth where quantities double over consistent time periods.
Logarithmic functions are the inverses of exponential functions. Problems like log₅ 25 < log₅ x require understanding logarithmic properties. These functions appear in applications like earthquake measurements and carbon dating.
💡 Quick Tip To easily identify function types, look at their position in the equation rational functions have variables in fractions, exponential functions have variables in exponents, and logarithmic functions have variables inside logarithms.
When solving exponential equations like 3ˣ = 32, apply the properties of exponents to find x = 5. For logarithmic equations like log₂x = log₂ 16, use the one-to-one property – if the bases are equal, then the arguments must be equal, giving x = 16.
Exponential inequalities require special attention to the base. When the base is between 0 and 1 (like in 0.5ˣ < 0.5³), the inequality sign flips when solving, giving x > 3. For logarithmic inequalities such as log₀.₃ x > 2, remember that the domain must be positive, resulting in 0 < x < 0.3².
Understanding domains and ranges is crucial. Exponential functions have domains of all real numbers but ranges of only positive real numbers. Logarithmic functions have domains of positive real numbers and ranges of all real numbers.
🔑 Remember The domain of a function becomes the range of its inverse, and vice versa. This property is essential when working with inverse functions!
Zero points are where functions cross the x-axis, and different function types have different patterns. Logarithmic functions usually have exactly one zero, while exponential functions of the form f(x) = bˣ typically have no zeros since they never cross the x-axis.
Rational functions display interesting behaviors determined by their numerator and denominator. In a function like f(x) = , the number of zeros equals the number of distinct factors in the numerator (four in this case), while the number of asymptotes relates to the denominator factors (three vertical asymptotes here).
Piecewise functions behave differently in different regions. For f(x) = , we apply the appropriate formula based on the input value. For x = 5, since 5 ≥ 0, we use the second piece, resulting in f(5) = 5.
📈 Visual Insight Think of rational functions as having "holes" (at asymptotes) and exponential functions as curves that never touch the x-axis but always approach it. These visual patterns can help you sketch graphs quickly!
When working with functions, you'll perform various operations. If f(p) = 5 and g(p) = -4, then (p) = 1, (p) = 9, (p) = -9, and (fg)(p) = -20. Understanding which operations yield maximum or minimum values is important for optimization problems.
Inverse functions undo the original function's effect. For f(x) = x + 5, the inverse is f⁻¹(x) = x - 5 because adding 5 is undone by subtracting 5. When finding inverses of complex functions, use the composition property f⁻¹(f(x)) = x.
Function composition combines functions, like (f ∘ g)(x) = f(g(x)). If (f ∘ g)(x) = 5x and g(x) = 3 - x, you can solve for f(x) by substituting f = 5x, so f(y) = 5 = 15 - 5y.
🧩 Function Tip When dealing with complex function relationships, identify what each piece does. If (f ∘ m)(x) = x, then m must be the inverse of f, since applying a function and then its inverse returns the original input!
Exponential and logarithmic functions shine in real-world scenarios. For chemical contamination with a half-life of 6 hours, we can determine when chicken feed is safe by using the formula (0.5)ᵗ/⁶ = 0.125, where t is time. After solving, we find that 6 × log₀.₅ 0.125 hours must pass.
Identifying functions from data tables requires examining patterns. In a table where doubling the input adds 0.5 to the output, we're likely dealing with a logarithmic function like f(x) = log₄ x. Finding the inverse function means swapping the x and f(x) values.
Growth models are powerful applications of exponential functions. If an outbreak doubles daily and reaches 1000 cases on the 31st day, then it reached 500 cases just one day earlier (the 30th). To calculate future growth, use the formula 1000 × 2ⁿ where n is the number of days after the 31st. By the third day of the next month, cases would reach 8000, exceeding the 5000-patient capacity.
🌍 Real-Life Connection Many natural phenomena follow exponential patterns—population growth, compound interest, radioactive decay. Recognizing these patterns allows you to make predictions about everything from investments to epidemic spread!
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.
4
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
bossing
@coteeee
Welcome to your guide on Rational, Exponential, and Logarithmic Functions! These powerful mathematical concepts appear in everything from population growth to chemical decay. We'll break down how to recognize, solve, and apply these functions in real-world scenarios, making these sometimes... Show more
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When tackling mathematics problems, being able to identify different function types is essential. Rational functions involve fractions with variables, such as . They often appear when calculating how resources relate to tasks.
Exponential functions have variables in the exponent position and model rapid growth scenarios like bacteria multiplication. These functions always have a positive range and appear in real-world applications like population growth where quantities double over consistent time periods.
Logarithmic functions are the inverses of exponential functions. Problems like log₅ 25 < log₅ x require understanding logarithmic properties. These functions appear in applications like earthquake measurements and carbon dating.
💡 Quick Tip: To easily identify function types, look at their position in the equation: rational functions have variables in fractions, exponential functions have variables in exponents, and logarithmic functions have variables inside logarithms.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When solving exponential equations like 3ˣ = 32, apply the properties of exponents to find x = 5. For logarithmic equations like log₂x = log₂ 16, use the one-to-one property – if the bases are equal, then the arguments must be equal, giving x = 16.
Exponential inequalities require special attention to the base. When the base is between 0 and 1 (like in 0.5ˣ < 0.5³), the inequality sign flips when solving, giving x > 3. For logarithmic inequalities such as log₀.₃ x > 2, remember that the domain must be positive, resulting in 0 < x < 0.3².
Understanding domains and ranges is crucial. Exponential functions have domains of all real numbers but ranges of only positive real numbers. Logarithmic functions have domains of positive real numbers and ranges of all real numbers.
🔑 Remember: The domain of a function becomes the range of its inverse, and vice versa. This property is essential when working with inverse functions!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Zero points are where functions cross the x-axis, and different function types have different patterns. Logarithmic functions usually have exactly one zero, while exponential functions of the form f(x) = bˣ typically have no zeros since they never cross the x-axis.
Rational functions display interesting behaviors determined by their numerator and denominator. In a function like f(x) = , the number of zeros equals the number of distinct factors in the numerator (four in this case), while the number of asymptotes relates to the denominator factors (three vertical asymptotes here).
Piecewise functions behave differently in different regions. For f(x) = , we apply the appropriate formula based on the input value. For x = 5, since 5 ≥ 0, we use the second piece, resulting in f(5) = 5.
📈 Visual Insight: Think of rational functions as having "holes" (at asymptotes) and exponential functions as curves that never touch the x-axis but always approach it. These visual patterns can help you sketch graphs quickly!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When working with functions, you'll perform various operations. If f(p) = 5 and g(p) = -4, then (p) = 1, (p) = 9, (p) = -9, and (fg)(p) = -20. Understanding which operations yield maximum or minimum values is important for optimization problems.
Inverse functions undo the original function's effect. For f(x) = x + 5, the inverse is f⁻¹(x) = x - 5 because adding 5 is undone by subtracting 5. When finding inverses of complex functions, use the composition property f⁻¹(f(x)) = x.
Function composition combines functions, like (f ∘ g)(x) = f(g(x)). If (f ∘ g)(x) = 5x and g(x) = 3 - x, you can solve for f(x) by substituting: f = 5x, so f(y) = 5 = 15 - 5y.
🧩 Function Tip: When dealing with complex function relationships, identify what each piece does. If (f ∘ m)(x) = x, then m must be the inverse of f, since applying a function and then its inverse returns the original input!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Exponential and logarithmic functions shine in real-world scenarios. For chemical contamination with a half-life of 6 hours, we can determine when chicken feed is safe by using the formula (0.5)ᵗ/⁶ = 0.125, where t is time. After solving, we find that 6 × log₀.₅ 0.125 hours must pass.
Identifying functions from data tables requires examining patterns. In a table where doubling the input adds 0.5 to the output, we're likely dealing with a logarithmic function like f(x) = log₄ x. Finding the inverse function means swapping the x and f(x) values.
Growth models are powerful applications of exponential functions. If an outbreak doubles daily and reaches 1000 cases on the 31st day, then it reached 500 cases just one day earlier (the 30th). To calculate future growth, use the formula 1000 × 2ⁿ where n is the number of days after the 31st. By the third day of the next month, cases would reach 8000, exceeding the 5000-patient capacity.
🌍 Real-Life Connection: Many natural phenomena follow exponential patterns—population growth, compound interest, radioactive decay. Recognizing these patterns allows you to make predictions about everything from investments to epidemic spread!
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.
4
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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