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Basic Function Types
The constant function f(x) = b always outputs the same value regardless of input. It creates a horizontal line on a graph with domain (-∞,∞) and range that's just the single value {b}.
The identity function f(x) = x returns exactly what you put in. Its graph is a straight line through the origin with domain and range both (-∞,∞). This simple function is surprisingly useful in many mathematical contexts.
Power functions show what happens when you raise x to different powers. The square function f(x) = x² creates a U-shaped curve called a parabola with domain (-∞,∞) and range [0,∞). Notice that this function never outputs negative values! Meanwhile, the cubic function f(x) = x³ has a distinctive S-shape with domain and range both (-∞,∞).
Try This! Graph f(x) = x² and f(x) = -x² on the same coordinate plane. How does the negative coefficient change the parabola's direction?
The square root function f(x) = √x starts at the origin and curves upward, with domain [0,∞) and range [0,∞). This function is the inverse of x² for positive values, meaning it "undoes" the squaring operation.
More Function Building Blocks
The cube root function f(x) = ∛x lets you work with roots beyond square roots. Unlike the square root function, it accepts negative inputs and produces negative outputs for them. Both its domain and range are (-∞,∞).
The reciprocal function f(x) = 1/x creates hyperbolas that never cross the axes. Since division by zero is undefined, x can't equal zero, giving this function a domain of (-∞,0)∪(0,∞). Its range is also (-∞,0)∪(0,∞).
The squared reciprocal function f(x) = 1/x² combines the reciprocal and squaring operations. Like the regular reciprocal function, it's undefined at x = 0, but since all outputs are squared, its range is [0,∞).
Important! Functions with asymptotes (like reciprocal functions) approach but never touch certain lines. This creates distinctive graph behaviors that are crucial for understanding limits in calculus.
The absolute value function f(x) = |x| outputs the distance from zero on the number line. It creates a V-shaped graph with domain (-∞,∞) and range [0,∞). The function turns negative inputs into positive values while leaving positive inputs unchanged.
Exponential functions like f(x) = aˣ or f(x) = eᵏˣ grow dramatically with positive inputs. When a > 1 or k > 0, they increase rapidly; when 0 < a < 1 or k < 0, they decrease toward zero. Their domain is (-∞,∞) and range is (0,∞).
Logarithmic and Trigonometric Functions
The natural logarithmic function f(x) = ln x is the inverse of eˣ. It grows very slowly as x increases, with domain (0,∞) and range (-∞,∞). This function is essential for solving exponential equations and modeling phenomena with diminishing returns.
The sine function f(x) = sin x creates a smooth wave that repeats every 2π units. This periodic function has a domain of (-∞,∞) and range restricted to [-1,1]. Sine is fundamental for modeling oscillations and cycles in nature.
Similarly, the cosine function f(x) = cos x also creates a wave pattern that repeats every 2π units. It's identical to sine but shifted horizontally, with domain (-∞,∞) and range [-1,1]. Cosine and sine work together to describe circular motion.
Visualization Tip: The sine and cosine functions can be viewed as tracking the y-coordinate and x-coordinate (respectively) of a point moving around the unit circle!
The tangent function f(x) = tan x is defined as sin x/cos x. It has vertical asymptotes wherever cos x = 0 . Its domain excludes these points, and its range is all real numbers (-∞,∞).
The secant function f(x) = sec x equals 1/cos x. Like tangent, it has vertical asymptotes where cos x = 0. Its range is (-∞,-1]∪[1,∞), meaning it never outputs values between -1 and 1.
Inverse Trigonometric Functions
The inverse sine function f(x) = sin⁻¹x (or arcsin x) reverses what the sine function does. Since sine outputs values only between -1 and 1, the inverse sine function only accepts inputs in this range. Its output is the angle (in radians) whose sine equals the input, with range [-π/2, π/2].
The inverse cosine function f(x) = cos⁻¹x (or arccos x) works similarly but finds the angle whose cosine equals the input. Its domain is [-1,1] and its range is [0,π]. This more restricted range compared to inverse sine helps avoid ambiguity since cosine is even.
Remember: While sin(sin⁻¹x) always equals x, sin⁻¹(sin x) doesn't always equal x—it only works for angles between -π/2 and π/2!
The inverse tangent function f(x) = tan⁻¹x (or arctan x) finds the angle whose tangent equals the input. Unlike the other inverse trig functions, it accepts any real number as input . However, its range is limited to (-π/2, π/2), representing just one period of tangent's repeating pattern.
The relationship between inverse trig functions creates useful identities. For example, sec⁻¹x = cos⁻¹, connecting the inverse secant directly to the inverse cosine function.
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Stefan S
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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
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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
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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❗️❗️⚠️
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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
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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
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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 Knowunity AI. 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 Knowunity AI. 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
Understanding Advanced Functions
Functions are mathematical relationships that show how one value relates to another. This collection presents common functions you'll encounter in algebra, precalculus, and beyond, along with their key properties and graphs. Understanding these functions gives you powerful tools to solve... Show more
Basic Function Types
The constant function f(x) = b always outputs the same value regardless of input. It creates a horizontal line on a graph with domain (-∞,∞) and range that's just the single value {b}.
The identity function f(x) = x returns exactly what you put in. Its graph is a straight line through the origin with domain and range both (-∞,∞). This simple function is surprisingly useful in many mathematical contexts.
Power functions show what happens when you raise x to different powers. The square function f(x) = x² creates a U-shaped curve called a parabola with domain (-∞,∞) and range [0,∞). Notice that this function never outputs negative values! Meanwhile, the cubic function f(x) = x³ has a distinctive S-shape with domain and range both (-∞,∞).
Try This! Graph f(x) = x² and f(x) = -x² on the same coordinate plane. How does the negative coefficient change the parabola's direction?
The square root function f(x) = √x starts at the origin and curves upward, with domain [0,∞) and range [0,∞). This function is the inverse of x² for positive values, meaning it "undoes" the squaring operation.
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More Function Building Blocks
The cube root function f(x) = ∛x lets you work with roots beyond square roots. Unlike the square root function, it accepts negative inputs and produces negative outputs for them. Both its domain and range are (-∞,∞).
The reciprocal function f(x) = 1/x creates hyperbolas that never cross the axes. Since division by zero is undefined, x can't equal zero, giving this function a domain of (-∞,0)∪(0,∞). Its range is also (-∞,0)∪(0,∞).
The squared reciprocal function f(x) = 1/x² combines the reciprocal and squaring operations. Like the regular reciprocal function, it's undefined at x = 0, but since all outputs are squared, its range is [0,∞).
Important! Functions with asymptotes (like reciprocal functions) approach but never touch certain lines. This creates distinctive graph behaviors that are crucial for understanding limits in calculus.
The absolute value function f(x) = |x| outputs the distance from zero on the number line. It creates a V-shaped graph with domain (-∞,∞) and range [0,∞). The function turns negative inputs into positive values while leaving positive inputs unchanged.
Exponential functions like f(x) = aˣ or f(x) = eᵏˣ grow dramatically with positive inputs. When a > 1 or k > 0, they increase rapidly; when 0 < a < 1 or k < 0, they decrease toward zero. Their domain is (-∞,∞) and range is (0,∞).
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Logarithmic and Trigonometric Functions
The natural logarithmic function f(x) = ln x is the inverse of eˣ. It grows very slowly as x increases, with domain (0,∞) and range (-∞,∞). This function is essential for solving exponential equations and modeling phenomena with diminishing returns.
The sine function f(x) = sin x creates a smooth wave that repeats every 2π units. This periodic function has a domain of (-∞,∞) and range restricted to [-1,1]. Sine is fundamental for modeling oscillations and cycles in nature.
Similarly, the cosine function f(x) = cos x also creates a wave pattern that repeats every 2π units. It's identical to sine but shifted horizontally, with domain (-∞,∞) and range [-1,1]. Cosine and sine work together to describe circular motion.
Visualization Tip: The sine and cosine functions can be viewed as tracking the y-coordinate and x-coordinate (respectively) of a point moving around the unit circle!
The tangent function f(x) = tan x is defined as sin x/cos x. It has vertical asymptotes wherever cos x = 0 . Its domain excludes these points, and its range is all real numbers (-∞,∞).
The secant function f(x) = sec x equals 1/cos x. Like tangent, it has vertical asymptotes where cos x = 0. Its range is (-∞,-1]∪[1,∞), meaning it never outputs values between -1 and 1.
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Inverse Trigonometric Functions
The inverse sine function f(x) = sin⁻¹x (or arcsin x) reverses what the sine function does. Since sine outputs values only between -1 and 1, the inverse sine function only accepts inputs in this range. Its output is the angle (in radians) whose sine equals the input, with range [-π/2, π/2].
The inverse cosine function f(x) = cos⁻¹x (or arccos x) works similarly but finds the angle whose cosine equals the input. Its domain is [-1,1] and its range is [0,π]. This more restricted range compared to inverse sine helps avoid ambiguity since cosine is even.
Remember: While sin(sin⁻¹x) always equals x, sin⁻¹(sin x) doesn't always equal x—it only works for angles between -π/2 and π/2!
The inverse tangent function f(x) = tan⁻¹x (or arctan x) finds the angle whose tangent equals the input. Unlike the other inverse trig functions, it accepts any real number as input . However, its range is limited to (-π/2, π/2), representing just one period of tangent's repeating pattern.
The relationship between inverse trig functions creates useful identities. For example, sec⁻¹x = cos⁻¹, connecting the inverse secant directly to the inverse cosine function.
We thought you’d never ask...
What is the Knowunity AI companion?
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.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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4.7/5
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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 Knowunity AI. 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 Knowunity AI. 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