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### Course: Algebra (all content) > Unit 7

Lesson 3: Functions and equations# Equations vs. functions

Equations and functions are not the same thing, but they can be related in several ways. Watch Jesse Roe and Sal talk about the difference between equations and functions. Created by Sal Khan.

## Want to join the conversation?

- if f(x) equals y then why bother use f(x).(29 votes)
- The following is from: http://www.purplemath.com/modules/fcnnot.htm

For functions, the two notations mean the exact same thing, but "f(x)" gives you more flexibility and more information. You used to say "y = 2x + 3; solve for y when x = –1". Now you say "f(x) = 2x + 3; find f(–1)" (pronounced as "f-of-x is 2x plus three; find f-of-negative-one"). You do exactly the same thing in either case: you plug in –1 for x, multiply by 2, and then add the 3, simplifying to get a final value of +1.

But function notation gives you greater flexibility than using just "y" for every formula. Your graphing calculator will list different functions as y1, y2, etc. In textbooks and when writing things out, we use names like f(x), g(x), h(x), s(t), etc. With this notation, you can now use more than one function at a time without confusing yourself or mixing up the formulas, wondering "Okay, which 'y' is this, anyway?" And the notation can be usefully explanatory: "A(r) = (pi)r2" indicates the area of a circle, while "C(r) = 2(pi)r" indicates the circumference. Both functions have the same plug-in variable (the "r"), but "A" reminds you that this is the formula for "area" and "C" reminds you that this is the formula for "circumference".(75 votes)

- What is a vertical line test ?(17 votes)
- A test to determine whether a relation is a function. When you graph a function (as Sal did at2:40), draw a vertical line at every point on the X axis (of course that's not practically possible, since there are infinite points, and besides, the whole paper would be black with lines - but do it in your head). If none of those vertical lines crosses the graph at more than one point, the relation is a function.

http://www.mathwords.com/v/v_assets/v11.gif(36 votes)

- arent linear equations technically functions?(9 votes)
- Most linear equations are functions. However, linear equations that create a vertical line would not be.(38 votes)

- when am I ever going to use this in my adulthood?(11 votes)
- How about when you go shopping and comparing prices, or when you a budgeting your money, paying bills, etc not to mention what applications it might have with your job.(20 votes)

- Instead of beating about the bush --A function has at least 2 variables: an output variable and one or more input variables. An equation states that two expressions are equal, and it may involve any number of variables (none, one, or more). A function can often be written as an equation, but not every equation is a function.(13 votes)
- FYI: Horizontal lines like y = 5 or f(x)=5 are functions but you only see the one variable in the equation. Thus, your first sentence is a little misleading. A function is a set ordered pairs where each input creates exactly one output. It does not need to be an equation. The equation for the functions does not need to have 2 variables. It just needs to be able to have / create ordered pairs that satisfy the rules for a function.(15 votes)

- Tell me if this question makes any sense: I noticed that using a graph was very affective in explaning what a function is. Can creating a graph resembling the one that was used in the video be the best way to solve it?(15 votes)
- yes making a graph to represent the one in the video is a good way to start out or at least that's what helped me(4 votes)

- What exactly is the difference between an equation and a function? As far as I could tell, Sal only gave examples of each and possible statements that could be made about equations or functions, but no clear definition as to what they are or the difference between them.(10 votes)
- A function is a set of ordered pairs where each input (x-value) relates to only one output (y-value). A function may or may not be an equation.

Equations are functions if they meet the definition of a function. But, there are equations that are not functions. For example, the equation of a circle is not a function.

This site might help: https://www.mathsisfun.com/sets/function.html(13 votes)

- what is one benefit of using functions?(5 votes)
- Functions have very many benefits, because functions have so many uses. As you learn more advanced forms of mathematics, you will find that functions can be used to simplify a concept or a statement. For example, 2x + 3 = y

One can say that a f(x), or a function of x, = y. So you can rewrite that equation as f(x) = 2x + 3. Now you can substitute "x" for any number you like. f(1) = 2(1) + 3. f(2) = 2(2) + 3. Hope this helped! Happy functioning!(7 votes)

- Aren't all functions also equations since functions contain an = sign (e.g f(x)=x+2x). An answer would be appreciated and I wish everyone a great day.(5 votes)
- Many functions are equations. But, they don't have to be. If you have a set of ordered pairs where each x-value relates to only one y-value, then you have a function.

For example: { (2,5); (3,8); (5,7); (-3,6) } is a function.(7 votes)

- Would f(x)=i^x be a function?(5 votes)
- A function is any expression that takes a variable or variables as input and produces an output, based on the function's operation. Assuming i is the square root of -1, then f(x)=i^x is a valid function, although the function may not have a value on the real number line at certain x values.(4 votes)

## Video transcript

SALMAN KHAN: I'm here with
Jesse Roe of Summit Prep. What classes do you teach? JESSE ROE: I teach algebra,
geometry, and algebra II. SALMAN KHAN: And now
you're with us, luckily, for the summer, doing
a whole bunch of stuff as a teaching fellow. JESSE ROE: Yeah, as
a teaching fellow I've been helping with
organizing and developing new content, mostly on the
exercise side of the site. SALMAN KHAN: And the reason
why we're doing this right now is you had some very
interesting ideas or questions. JESSE ROE: Yeah, so
as an algebra teacher, when I introduce that concept
of algebra to students, I get a lot of questions. One of those
questions is, what's the difference between an
equation and a function? SALMAN KHAN: The difference
between an equation verses a function, that's an
interesting question. Let's pause it and
let the viewers try to think about
it a little bit. And then maybe we'll
give a stab at it. JESSE ROE: Sounds great. So Sal, how would you
answer this question? What's the difference between
an equation and a function? SALMAN KHAN: Let me think
about it a little bit. So let me think. I think there's
probably equations that are not functions
and functions that are not equations. And then there are probably
things that are both. So let me think of it that way. So I'm going to draw-- if
this is the world of equations right over here, so
this is equations. And then over here is
the world of functions. That's the world of functions. I do think there
is some overlap. We'll think it through
where the overlap is, the world of functions. So an equation that is not a
function that's sitting out here, a simple one would
be something like x plus 3 is equal to 10. I'm not explicitly talking
about inputs and outputs or relationship
between variables. I'm just stating an equivalence. The expression x plus
3 is equal to 10. So this, I think, traditionally
would just be an equation, would not be a function. Functions essentially
talk about relationships between variables. You get one or more
input variables, and we'll give you only
one output variable. I'll put value. And you can define a function. And I'll do that in a second. You could define a
function as an equation, but you can define a function
a whole bunch of ways. You can visually
define a function, maybe as a graph-- so
something like this. And maybe I actually
mark off the values. So that's 1, 2, 3. Those are the
potential x values. And then on the
vertical axis, I show what the value of my
function is going to be, literally my function of x. And maybe that is 1, 2, 3. And maybe this
function is defined for all non-negative values. So this is 0 of x. And so let me just draw--
so this right over here, at least for what I've drawn
so far, defines that function. I didn't even have
to use an equal sign. If x is 2, at least the way
I drew it, y is equal to 3. You give me that input. I gave you the value
of only one output. So that would be a legitimate
function definition. Another function
definition would be very similar to what you
do in a computer program, something like, let's say, that
you input the day of the week. And if day is equal to Monday,
maybe you output cereal. So that's what we're
going to eat that day. And otherwise, you
output meatloaf. So this would also
be a function. We only have one output. For any one day of
the week, we can only tell you cereal or meatloaf. There's no days where you
are eating both cereal and meatloaf, which
sounds repulsive. And then if I were to
think about something that could be an
equation or a function, I guess the way I think about
it is an equation is something that could be used
to define a function. So for example, we could say
that y is equal to 4x minus 10. This is a potential
definition for defining y as a function of x. You give me any value of x. Then I can find the
corresponding value of y. So this is at least how
I would think about it.