Difference between equations and functions | Functions and their graphs | Algebra II | Khan Academy

Jesse Roe and Sal talk about the difference between equations and functions

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[bluedragonrob]

Thanks guys. I dont understand everything, but ill never stop trying to learn.

[alcesmir]

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.

[orlrogc]

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.

[ncroc]

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

[GetMeThere]

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.

[raygorish]

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.

[patelvidhu]

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.

[drewpasttenseofdraw]

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.

[brenoakiy]

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...

[Dosidis]

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,

[mathsbynd]

Some reference say "functions are subset of a a equations". How about it?

[BirendraKumar-nrnb]

Is it like equations with two variables both function and equation when one of the variable is on the other side of equal(=) ???

[visiting]

what's the difference between differentiating a function and an equation?

[jasonakon]

Cool. Thanks for the info and knowledge

[MadaxeMunkeee]

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..

[saikatdey]

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

[TheKomentor]

"Download this video" in the description is available for other YouTube videos?

[moamenabdelmageed]

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?

[starstalker_awe]

"That souds repulsive" in the undertone voice. Love it

[alexchan]

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.