Difference between equations and functions | Functions and their graphs | Algebra II | Khan Academy
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- Опубликовано: 24 июл 2012
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Jesse Roe and Sal talk about the difference between equations and functions
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Thanks guys. I dont understand everything, but ill never stop trying to learn.
This is gold
That is past 70,000 years of history right there!!
The f(x) is not only fancier, but also more powerful and more convenient.
For example, f(x) = x² is easily written as y = x². However, f(x+3) = (x+3)² is a notion that can't easily be expressed with just y.
By definition, a function has only one output for any given input(s). That is the basis for the vertical line test.
Integer division and taking the modulus are distinct operations. That is why in a computer program you would write 10/3 or 10%3 depending on the desired result. As humans we write 3R1 only for the sake of being succinct.
In matrix algebra, a matrix or vector (regardless of size) is considered a single variable. So A x B = C is valid because C is the single output.
Cool. Thanks for the info and knowledge
Great question and something we're just getting into again. To the first example you gave of an equation, which is not a function, if you were to solve & graph that equation, as x=7, it would not qualify as a function. I thought that might be a way I could extend it. Also, the equation of a circle would also be a great example to put in that section. It has what looks like an input & output(x&y) in some standardish form, but it's not. It's important for students to see x & y in non-functional equations, I think. I like the venn diagram application. I'd add a statement of equality is just one of the ways we may use to represent functions, so in addition to your graph and meatloaf/cereal example, lists of ordered pairs and a table would be other common ways to express functions, which are not equations. Great video and food for thought, Khan Academy. I know what my class is doing next week.
You can learn how equality relation and function is defined by looking at an abstract algebra book. They are both subset of cartesian product of two sets S1xS2 where for function you need 1-1 and onto mapping from f:S1-> S2, (x,f(x)) in S1xS2. For equavalence relations, you need reflexitivity, symmetry and transitivity. So if you wanna learn more, use keywords above to do google search
"That souds repulsive" in the undertone voice. Love it
I still need to understand more about WHY math is interested so much in functions, and needed to highly define and specify what functions are, etc.
I want to hear more about the real BACKGROUND theory of these things.
"Download this video" in the description is available for other RUclips videos?
Salman, you should do a video on Einsteins General and Special Theory Relativity. That would be awesome!
Long story short: an equation is a constant function
A function is something that gives you number, when you give it an appropriate input. An equation is relation between variable and numbers and there probably one or several unknown quantities in it.
great video! :) keep 'em coming!
A function is a rule that establishes a mapping or relationship between inputs and outputs.
If f(x), where f(x) is some rule, for instance: x^2 or sqrt(x+2) or (x-3)/(x+2) or etc
assigns each member in the domain X to a unique member in the range Y the f(x) is a function.
Nice explanation!
Thank you!
A function by definition has one output per input. A relation has at least 2. The "vertical line test" is just an easy way to check how many outputs you have per input.
Thanks for everything!
the quadratic function is not the quadratic equation, these are two different things so I think a function contains input and output but an equation does not contain input and output.
I think the most distinguishing property of an equation is that it is a logical statement that serves as a proposition in an argument. Functions, that satisfy the definition are still fundamentally different to equations in that we do not assign a truth value to a function. Rather, it is either well defined or it isn't..
Differentiating a function means finding the derivative (which is a measure of how a function changes as its input changes.)
When you are defferentating an equation, you are actually considering each side as a function and you are using the theorem that states f(x)=g(x) => f'(x)=g'(x) .
Note that the inverse theorem is not valid,
Converse (as opposed to inverse) is not true.
Functions are all over the place (physics, economics, biology...)
If you were to develop mathematics, especially applied mathematics, you would inevitably come across functions.
Examples are force as a function of mass and acceleration F(m,a)=m*a, cost as a function of price and quantity c(p,x)=p*x, pressure of a gas as a function of volume, temperature and number of moles P(n,T,V)=n*R*T/V and so on...
In short,
|x| is fun y=|x| is equation
X^2+y^2 is a fun but x^2+y^2=0 is eqn
Love from India
What book do you use for algebra 2
thanks for upload :D
what's the difference between differentiating a function and an equation?
Differentiation is a procedure for translating all output values of a given function into a another set of values. IOW it produces another, different function. As to the difference between a function and an equation, well, that is the purpose of this video, although I think he answers it poorly.
thanks
I'd have loved to hear Roe's take on it aswell.
So what about multi-valued functions? Those exist aswell, after all...
Of course, you can usually split them up into single-valued functions, but still...
An example would be integer division + modulo.
10/3 = 3R1 - that would be a function with two output variables.
Or what about a cross product? 2x3-variable input, 3-variable output.
Of course, you could say that's an operator but operators and functions are pretty much the same thing anyways...
Yes, you can SOMETIMES isolate some variable and make it as a function of others.
Normally when you work in physics, you try to make all variables in a function constant except for one, so you get something like f(x), a function of one variable, in order to study some phenomenon. (in the previous gas law example, I could set a value for n and V, so the function would become P(T)=C*T, where C(constant)=n*R/V)
Something deeper?
Pretty much all of current physics are described by functions...
Is it like equations with two variables both function and equation when one of the variable is on the other side of equal(=) ???
Actually, it depends how you define it. It could mean any function of x. So it isn't neccessarily y.
So basically: a function describes a relation between variables and an equation is a function with a specific value for the variables (which are unknown constants now)?
Do u mean that: for an equation to be the same as the function, we must have only the variable Y alone on one side of the equation, and all other variables and values to be on the other side?
Some reference say "functions are subset of a a equations". How about it?
Thanks i already knew how to get the answers right on a test involving f(x) i just didnt know why it was written that way, i think i get it now tell me if this is right. When you say y=x+1 you would have to make an assumption about what needs to be done in order to provide a solution. However if you put f(x)=x+1 it is explicit because there is only one thing that can be done with this equation.
I'm not sure if they've post edited this video but it seems odd that in the video and the video description they introduce Jesse Roe but the only interaction with Jesse and Sal is Jesse is asked a question and Sal answers it fully without a retort from Jesse. It's like, What's the point in having him in it at all? I feel they might have edited this video in RUclips's editor for curation simplification purposes but if that's the case then why not recast the whole video? Very bizarre.
How to contact Sal ?
You guys are doing a fantastic job. On the difference between a function and an equation, I think you are missing the fundamental point.
A function is a map from one set to another. An equation is a logical statement defining a solution (space).
The fact that the solution set of an equation could be identical to a function map is a fun curiosity, and certainly not a good pedagogical illustration of the two concepts.
Right. But you really only supply examples. And they're always equations in more than one variable--as soon as you have more than one variable, you can say one variable gets to be a function of all the others. Equations I understand, but I don't understand why it seems so CRITICAL in mathematics to carefully describe FUNCTIONS.
It must relate to something deeper (for example, calculus seems to be especially concerned with functions).
I'm quite sure I'm not the only one who asks such questions.
Well, f(x) means function of x, kind of like the same thing as y.
So your saying that sets can be an output of a function. Because it's still a single set C or whatever? Beast.
yes
If a function is an equation, the quadratic function will be the quadratic equation. However, the quadratic function is not the quadratic equation thus a function is not an equation.
And there is no difference between differentiating a function and an equation, provided that your function/equation is differentiable.
This video began with an actual math teacher who apparently had some interesting ideas... but then Sal runs off and the original share is never completed... Sigh...
Dafuck I still don't get it
equations don't have to have multiple variables, whereas functions do (think of it as the value of the one "functions" off the other).
Same
can someone explain what f(x) means.
It is read as function of x. It means we are talking about a function which has the input variable x. Remember that a function is where you input one value and get another. The input is usually referred to as the variable x and the output as f(x). So give me a formula that gives the mapping of each value of x to another value and we use f(x) to represent the output -- but not for any single value of x. The formula describes the rule for translating x into ANY value of f(x).
Sorry to pick nits, but I am a languages person studying math, and at 0:36 the subtitles should read "versus" rather than "verses."
He seems to be winging it.
Ok, I guess that's fair enough.
who's here from 2021
its y
were you guys using discord to call?
:D
2 views :/
This academy is good but the color can not accommodate everyone
meatloaf
If a function is an equation, the quadratic function will be the quadratic equation. However, the quadratic function is not the quadratic equation thus a function is not an equation. Therefore your teaching is incorrect.
Yes i agree with you every equation is a function but vice versa is not true take an example x+7=0 , x=y^2 , |y|=x-1
These are equation not function
So every function you can say it's an equation .
I give up
Gando.