What is the difference between a function and an equation?

[Momto6inIN]

I love Video Text, but functions are the one thing it does not adequately cover. So I get that f(x) essentially can be considered the same as y. But why the sudden change in notation? Googling basically says that a function is a little different than an equation, but I didn't follow why.

[vonfirmath]

When you have an equation
(Generally something like 4x-6=0) Then you are trying to find values of X that make it equal 0.

IF it is 7y+6=4x-6 --> Then again, you are looking for the specific point that the two equations are the same.

 

A function allows you to put multiple values in and get multiple values out:

f(x)=4x-6

 

Edited August 23, 2019 by vonfirmath

[LJPPKGFGSC]

A function is a special type of relationship in math where each input has exactly one output.  There are several ways to write a function.  You can draw a graph.  You can organize the inputs and outputs into a table.  You can draw a mapping.  A mapping is like a list, but you draw arrows to show the relationship from each input to the correct output.  If you look at the Wikipedia page linked below, the mappings are on the right side of the page, with big ovals marked, X and Y.  There is an oval for Set X, which is the list of all elements that are inputs.  This is called the domain.  And there is Set Y, which is the list of all the items that are outputs, which is called the range.

So, an equation is one more way to explain (define) a function.  It is often the most useful way.  You would not want to make a list (either a table or a mapping) of a function where any number can be used as an input.  You cannot write an infinitely long table.  But you can write an equation, which is valid for all real numbers as the input.

Does that help? 

(But this Wikipedia page is way too detailed if you are just beginning to study functions. I only linked it because you mentioned it and it has some good diagrams.

https://en.wikipedia.org/wiki/Function_(mathematics)

Edited August 24, 2019 by LJPPKGFGSC
typo

[Jann in TX]

Another way to explain the difference using 2 variable problems...

x + y = 3 is an equation.  The = means that the left side and right side are balanced.  You can manipulate the equation into many different equivalent forms such as:

x - 3 = y,  y = 3 - x and y = -x + 3

You cannot solve this type of equation unless you know one of the x or the y values OR if you have another equation in the same system.

--

You can write the same equation as a function. A 'function' is a type of coding (think computer programing).  It is a streamlined way to show that you are interested in ALL possible combinations that make the equation true.  You can express the answer as ordered pairs and the equation will have a 'shape' (in this case a straight line).   In the case of the original problem y is usually considered the 'result' variable so writing the equation in function notation would be:

f(x)= -x + 3  or f(x) = 3 - x

The f is the name of the function-- since the word 'function' begins with an f it is the most common name.  If your equation was all about finding the distance if you plug in a value for x then your coding COULD look like:

d(x)= -x + 3    <-- d is for distance!

You can interpret this to be "You can find the distance (d) if you plug in a number for (x) into the following (=) expression  (-x + 3)

--

When I teach beginning functions I personify the problem:

f(x) = 2x -5  would be "Frank is holding up a sign that says '2x - 5'

f(3) = 2x - 5 would be Frank is holding up a sign that says '2x - 5'.  How much would his sign be worth if you plugged a 3 in for the x?  answer would be:

f(3)= 2(3) -5  so  f(3)= 1

--

g(a)= a + 7 would be

George is holding up a sign that says ' a + 7' 

g(5) = a + 7 would be 'how much would George's sign be worth if you plugged in the 5 for the a...

g(5)= 12

SOOOO  the first letter in the function code is the NAME (often big problems will have many equations/functions interacting so we name the parts) the letter in the first parenthesis is the variable you can substitute a number for--OR if there is a number inside the (  ) it is telling you to use that number and substitute it for the variable in part after the =.  The = is more of an arrow that points to the expression (sign frank is holding up).  The math expression that has no value unless you switch out the variable for a number.

For big problems that contain several functions (usually having a common variable) it is often easier to combine the expressions (signs they are holding up) and THEN substitute a value for the variable into ONE problem instead of substituting the value for the variable in all of the functions separately then combining the functions (this is especially true when your functions contain a lot of exponents!).

 

 

[letsplaymath]

On 8/23/2019 at 9:14 AM, Momto6inIN said:

So I get that f(x) essentially can be considered the same as y. But why the sudden change in notation?

A change in notation often signals an increase in abstraction. Along with that increase in abstraction comes the power to talk about whole new classes of number relationships.

For example, in elementary school, we deal with just-plain-number equations. Simple, single relationships, like 2+3=5.

In middle school, we step up the ladder of abstraction to deal with a whole class of number relationships all at once. We are not really interested in specific numbers, but on broader relationships between numbers. The equation "x+3=y" gives us a collection of numbers that are all related to each other because "This is three more than that."

In high school, we step up the ladder again. Now we are less interested in any particular equation. Instead, we are looking at whole classes of equation-relationships. Not just the one equation x+3=y, but all of the x+n=y type of functions. Or even more broadly, any function where we take in an x and output a related value f(x).

In this new level of abstraction, often we are not trying to solve a particular equation. Instead we are looking at what a whole class of equation-relationships has in common, and how they are different from this other class of equation-relationships. Or at what happens when we combine functions --- put a value into one function and then take the output and dump it into another function --- and does it matter which order we do the functions? Or can we find a way to go backwards --- if we know the output, can we figure out what the original input would have been?

In the early days of working with functions, it can seem like not much has changed except the notation. And so it can seem like a ridiculous thing to do. Why change something that's not broken?

But as the student moves on, the power of the new notation will become more important because it gives them a way to think about bigger ideas.

[mathnerd]

In a function, there are a number of inputs and a corresponding output for each input. 

also, there exists something called an "inverse of a function" which can basically do the opposite (i.e. undo the effects) of a function.

Edited August 25, 2019 by mathnerd

[Farrar]

I know the above are way better explanations with clearer rules and all, but with my kids I always harken back to the imagery of little kid math books. Equations are balance scales. Functions are the little machines in Anno's Math Games where you can feed in different numbers and get different results.

[Momto6inIN]

Thank you all, this is very helpful! I think all your responses will help me better help DD this week 🙂

[EKS]

The word function describes a certain type of relation, where each x value produces (or is paired with) only one y value so that if you were to graph the relation, it would pass the "vertical line test."  When you're talking about functions that can be written as equations, you can think about them as a specific type of equation, one where each x value produces only one y value.  Function notation--f(x), g(x), etc--is just a different way to write an equation that is a function. 

One advantage of function notation is that it allows you to say certain things in a more compact way.  For example, instead of saying "Find y when x is 3,"  you can say "find f(3)."  You can also show substitution--for example, "f(x+1)" or "f(g(x))." 

Function notation becomes more convenient as you move along in math,  but in the beginning it seems kind of clunky and odd and unnecessary, and the trend toward introducing it earlier in the math sequence, where it really is just acting as a synonym for y, is, IMO, misguided.

[Momto6inIN]

21 hours ago, EKS said:

The word function describes a certain type of relation, where each x value produces (or is paired with) only one y value so that if you were to graph the relation, it would pass the "vertical line test."  When you're talking about functions that can be written as equations, you can think about them as a specific type of equation, one where each x value produces only one y value.  Function notation--f(x), g(x), etc--is just a different way to write an equation that is a function. 

One advantage of function notation is that it allows you to say certain things in a more compact way.  For example, instead of saying "Find y when x is 3,"  you can say "find f(3)."  You can also show substitution--for example, "f(x+1)" or "f(g(x))." 

Function notation becomes more convenient as you move along in math,  but in the beginning it seems kind of clunky and odd and unnecessary, and the trend toward introducing it earlier in the math sequence, where it really is just acting as a synonym for y, is, IMO, misguided.

Re the bolded above - does this mean an equation like x squared + 17x + 70 = 0 is not a function? Because x can be 2 different values?

 

Edited August 28, 2019 by Momto6inIN
I just realized that's a dumb question as that isn't a line with a y. Can you give me an example that wouldn't be a function?

[EKS]

1 hour ago, Momto6inIN said:

Re the bolded above - does this mean an equation like x squared + 17x + 70 = 0 is not a function? Because x can be 2 different values?

What you have there is not a function because it doesn't have an "input" and an "output."  In other words, a function would have to have both an x and a y in it. 

But there is a related function, which is y = x^2 + 17x + 70.  When you solve your equation, you are answering the question, "in the function y = x^2 + 17x +70, what is x when y = 0?"  If you were to graph that function, you'd get a parabola that crosses the x-axis (where y = 0) in two places, at -10 and -7, and these are the solutions to your equation.

When you graph the related function y = x^2 + 17x + 70 you'll see that each each x value produces only one y value, and so it passes the vertical line test.  In other words, when we plug in -10 for x, we get just 0, not 0 and some other number, and if we were to do this for every single x value, the same thing would happen--the solution would be only one y value. 

Note that a relation can still be a function if the same y value (in this case 0) has two different x values (in this case -10 and -7).

[RootAnn]

But x= y^2 + 17y +70 would not be a function. If you graph it, it will be a parabola on its side (90 degrees off the previous equation).

The "function test" is often known as the vertical line test. (Several good tutorials on this.)

A circle would be an example of a not-a-function.

x² + y² + Dx + Ey + F = 0

That is a standard circle equation, but it is not a function.

Edited August 28, 2019 by RootAnn

[daijobu]

4 hours ago, RootAnn said:

But x= y^2 + 17y +70 would not be a function. If you graph it, it will be a parabola on its side (90 degrees off the previous equation).

 

 

But only if you intend to graph the function on the x-y plane.  One could use any "dummy variable" including y:  f(y) = y^2 + 17y + 70.  

One of my favorite Math With Bad Drawings is the Baby Name Book of Variables which advises against naming a function "x" and the variable "f":