# Function concept and equation concept

1. Sep 19, 2013

### thedy

Hi,I know,that my question is trivially and frequently asked question,but thread,where I have colud post was closed.So I made new one.
I can report,what I know.
Equation is something what equals each other.One statement to another.So 10=2x is equation.

Function is concept which gives us information about,better said relation between input and output.So y=2x is function,but equation as well.But why,for example 10=2x is not a function?It shows us relation between x and 10,doesnt it?
Please give me answer for my example.I am just beginner.And yes,I know that function is some kind of machine which takes something and gives us something.So in 10=2x we give 5,and get 10.

2. Sep 19, 2013

### thedy

I forget my one remark.It seems,like everything what have one unknown on left one others unknowns on right is function.If we have number on left side,it is not

3. Sep 19, 2013

### chiro

Hey thedy.

A function expresses a map in terms of some inputs that produce one output.

A function has to be explicit: this means that you have the output variable on one side (usually LHS) and the other variables on the other.

If this is not done, then you don't have a function.

A few examples of a function include:

y = x^2 + 3
z = xy + sqrt(y)*e^(-x)

An non-example is:

y + e^(-y^2) = xz + sin(z)cos(x) + e^(-xz)

In the above you can't get y by itself on one side and a function requires that you can do this.

You can still evaluate expressions like this, but they aren't strictly speaking functions.

So in summary, if the functions are defined and you can put the variable on one side (like x,y,z,etc) and the function can be evaluated at all points in the domain, then you have a function.

You also need a vertical line test meaning that if you draw a vertical line anywhere on the axis of the output variable, then you will only get one intersection with the function at that point. Any more than that and you don't have a real function.

4. Sep 19, 2013

### arildno

1. "Equation is something what equals each other.One statement to another.So 10=2x is equation."

a) Not exactly.
I would say that an EQUALITY "is something what equals each other", whereas an "equation" is a specific TYPE of an "equality".
b) To understand what sort of type of equality an equation is, it is helpful to understand another type of equalitythat we call IDENTITY.

Look at the expression x=x (where x is some number).

Now, ask yourself the question: For which x's is that expression TRUE?
Answer: For EVERY choice of "x", the statement x=x is a true statement! (Agreed? 2=2, 0.5=0.5, 123=123 and so on..)

Other identities are 2*x=x+x, 3x-x=2x and so on.

c) Now, we can tackle the meaning of an "equation"!
An equation is a type of equality that only holds true for SOME choices of x, but not for all choices of x.

For example:
10=2*x is a true statement ONLY if x=5, but not for any other choice of x we might want to insert on our right hand side. This distinguishes an equation from an identity. We call such x's that make an equation a true statement for the solutions of the equations.

x^2=9 is also an equation. Here, we have TWO solutions, x might be either of the two numbers -3 or 3, but NOT any other number!

Thus, equations can have multiple numbers as their possible solution, but only in identities are ALL numbers possible solutions.

5. Sep 19, 2013

### thedy

Thanks to you,gentlemen.Really clarifying answers.It gives me more light on it.I really appreciate,that you give me answers.However,as I know me,after better thinking over,I will have bizarre questions,again....:)
So,once again,thanks

6. Sep 26, 2013

### thedy

Well,I read it again.If I am right,then function is subgroup of equality?

7. Sep 26, 2013

### Staff: Mentor

No, not at all. A function is a pairing (or map) of elements of one set to another set (not necessarily different. The simplest kind of function pairs a single number in the domain with a single number in the co-domain.

The formula for a function is usually expressed as an equation, so maybe that's where you are confused.

Here's an example:

f: R → [2, ∞), such that f(x) = x2 + 2

The first part, f: R → [2, ∞), says that f, the function, pairs a real number with a number in the interval [2, ∞). Here the domain is all real numbers, but the co-domain is the real numbers greater than or equal to 2.

The second part, f(x) = x2 + 2, is the formula for the function, which you can use to find a function value (output) for a given value in the domain (input). For example, f(3) = 32 + 2 = 11.

The graph of the function is the set of ordered pairs (x, x2 + 2). One point on the graph is (3, 11). By plotting a number of points of the graph you can see that the graph is of a parabola whose vertex is at (0, 2).

8. Sep 26, 2013

### Staff: Mentor

An equation is a statement that two quantities are equal. In inequality is another type of statement, where two quantities are different in their values (i.e., not equal).

Yes, 10=2x is an equation, one that I would call a conditional equation, an equation that represents a true statement only under certain conditions. For this equation, the condition is that x = 5.

Another type of equation, which arildno mentioned, is an identity, an equation that is true for all values of whatever variable is present. For example, (x + 1)2 = x2 + 2x + 1 is true for every number x.
No, this equation doesn't represent a function. There is only one number (5) that you can put in to get 10. If you put in any other number, you don't get 10. If you had f(x) = 2x, that would be a function, where the domain and co-domain are implicitly all real numbers.

9. Sep 26, 2013

### economicsnerd

My take on it:

1) An equation is a sentence, written in the language of mathematics, in which the verb is "equals" (typically written as "$=$"). For instance, the sentence "$\text{blah blah blah} = \text{ yada yada}$" is an equation.

2) A function is a machine. When I say "$f:X\to Y$", what I mean is that $f$ is a machine which takes an input $x$ from $X$, and then spits out some output $y$ from $Y$. For a given choice of $x$, we sometimes label the output as $f(x)$. So the equation/sentence $y=f(x)$ gives a description of our function.

So when you see the equation/sentence "$y=x^2-3x +1$", the way to interpret the sentence is, "I have in mind a function, which takes a number $x$ as an input, and spits out $x^2-3x +1$ as an output."

Last edited: Sep 26, 2013
10. Sep 27, 2013

### thedy

Thanks. :) Yes,I understand,that function describes something different,than equation.But if we express function in form of equation,we can (in this case) function classify as some kind of equality.Function in form of equation said:,,Machine is equal to command".With command I mean content of function.F(x)=command,2x=content....
So in function we can have infinite options to have output,whereas in equation we have output only one,and fixed?

11. Sep 27, 2013

### jbriggs444

I am having trouble understanding this.

Yes, I would agree that for those functions that can be defined by an equation, there is a natural association and one can think of the equation and of the associated function and blur the distinction about which is which.

But there are equations that do not define functions and there are functions that cannot be defined by equations.

x2+y2 = 1 is an equation. But it does not define a function.

Absolute value is a function. But it is not definable by any single simple equation involving only addition, subtraction, multiplication and division.

Not all functions have infinitely many possible inputs. The "domain" of a function is set of all of its valid inputs. One can have functions whose domains are infinite, large or small. One can even consider the function whose domain is empty.

The truth value of an equation is a function of its free variables. So one might identify an equation with a boolean function. In this sense "x = y" could be identified with a function f in two variables such that f(x,y) is 1 if x is the same as y and is 0 otherwise.

12. Sep 27, 2013

### Staff: Mentor

I don't see how this is useful. With a function there is often (but not always) a formula that shows how function values are calculated, based on the input value. Here is an example:
f = {(1, 2), (2, 1), (3, 3)}.
From this we see that f(1) = 2, f(2) = 1, and f(3) = 3. There is no formula given, and in fact, no formula is necessary.

Neither of this is necessarily true. A function might have a domain that consists of only a small number of values.

An equation has two possible values - true or false -- depending on whether the equation is true or not.

Some equations are always true (identities), such as 2(x + 1) = 2x + 2.
Some equations are sometimes true and sometimes false (conditional equations), based on the value of a variable. An example is 2x + 3 = 5. This equation is true if x = 1, and is false for all other values.
Some equations are never true (contradictions), such as 2x = 2x + 1. There is no value of x that makes this a true statement.

13. Oct 1, 2013

### thedy

I am confused,but how I see function concept and equation is such,that function is some kind of instruction,how to get some number.And equation is only finding of variables.Function is something more general for me.But correct me,if I am wrong.
Next,I do not understand one thing.For example we have state equation pV=nRT.Now,this is equation,but what if we pV change with work?Now,we have W=nRT.Is it a function now?I do not explain better,what I mean,but it seems like every expresion we can change with name of the function.Also nRT can be labeled like function by some sign....But of course,we definetly do not get function from equation only by trivial exchange signs,but I want to know,why.
Thanks....I know,that you explained me everything,but how longer I read this thread,more confused I am.

14. Oct 1, 2013

### Staff: Mentor

If you exchange pV for W to get W = nRT, then yes, W is a function of n and T (as I recall from my chemistry long ago, R is a constant).
I don't understand what you're trying to say.
W = f(n, T) = nRT. Is that what you mean?

Your writing is not very clear, so I would guess that English is not your native language.
A function is a rule that pairs elements of one set with elements of possibly some other set. The rule is often, but not always, expressed by an equation. The concepts of function and equation have some ties, but they are different concepts.

15. Oct 1, 2013

### thedy

Thanks,yes I do really sorry for my English.It is not my native language and I am disturbed because of not understanding function and equation and I was writing this message too fast.But back to my issue.
I give other example.If we have y=mx+b and f(x)=mx+b,what is the difference?I guess,none.I just want to see difference.If function ,,is a rule that pairs elements of one set with elements of possibly some other set",than my question is why equation do not pairs elements.Or how I know if it does.

Or,if we assume that domain consists only of one number,that means range will be too only the one number,can we say,that it is still a function?Or it is equation then?

I do not really understand why 10=2x is not function.You said,that equation does not map element.What does it mean,in reality?You put 5,and get 10.I know,you said,that we have only one true statement.But if function is confined(domain is one number)then,what is the difference between this function and ordinary equation?You said,that equation is statement.But function is rule,which technically it is statement too.,,I state that this is rule".OK,I know,it is pointless,but I just want to show,that for me at least,it is not clear explanation.

So,sorry once again,I will be glad,if you answer me,if not I will understand it,because my questions are weird.
Because of to make it easier,why we can not consider ordinary equation to be a ,,machine" like function?

Last edited: Oct 1, 2013
16. Oct 1, 2013

### VRT

I'll approach the issue from the usability stand point. "A function is a map" means you can use and reuse it for different input parameters (as a function in computer library) and get results from it. For example, f(x)=3x^2+6. You can feed it with many real numbes from some interval (-10.0 to +10.0 for example) and draw points on a plane with X=x (input) and Y=f(x) (output). In the end you'll get a curve - you've used a function to draw it.

Equations cannot be used like that. They are to be solved and you'll get a set of solutions such that when you substitute the unknown (not input parameter!) with a solution, left and right sides become equal. Once solved, it's done. You cannot solve it again and get different result.

Now, a function can serve as a part of an equation - usually to simplify writing.

17. Oct 1, 2013

### Staff: Mentor

There's not much difference. In the first equation, y is a function of x (assuming that m and b are constants). The second equation uses function notation (f(x)), and uses the same formula as the first equation. Both equations give the rule that maps an input (x value) to an output (y value or value of f(x)).

Since there are no restrictions on the domain, we can assume that the domain is all of the real numbers.
Technically I suppose that we'd have to say that it's a function, but with a domain of only one number, it's not a very interesting one. I've never seen any textbooks that call an example such as this a function.

Your equation is equivalent to x = 5, which represents a single point on the number line.
The idea behind the "machine" analogy is that if you put different things into the machine, you get different output values. If the machine accepts only one single value, and from that, it always produces the same output value, it's not a very useful machine.

18. Oct 1, 2013

### Staff: Mentor

There's not much difference. In the first equation, y is a function of x (assuming that m and b are constants). The second equation uses function notation (f(x)), and uses the same formula as the first equation. Both equations give the rule that maps an input (x value) to an output (y value or value of f(x)).

Since there are no restrictions on the domain, we can assume that the domain is all of the real numbers.
Technically I suppose that we'd have to say that it's a function, but with a domain of only one number, it's not a very interesting one. I've never seen any textbooks that call an example such as this a function.

Your equation is equivalent to x = 5, which represents a single point on the number line.
The idea behind the "machine" analogy is that if you put different things into the machine, you get different output values. If the machine accepts only one single value, and from that, it always produces the same output value, it's not a very useful machine.

19. Oct 1, 2013

### arildno

It IS possible to connect the concept of a function with that of "equality", but in a rather tricky, nasty way:
Look at the "things" f(0), f(-1), f(2), f(0.33) and so on (you've got an infinite number of such!)
Then, if our expression for the function is, say, f(x)=2x, we can look upon that as a SHORTHAND for the INFINITE number of DEFINING equalities,
f(0)=0 (by definition of it)
f(-1)=-2 (by definition)
f(2)=4 (by definition)
f(0.33)=0.66 (by definition)

This is to write out the PAIRING rule you have been told about, on the one hand between your x's and your f(x)'s.

We call this type of pairing as a MAPPING, for a very good reason:
There is an infinite number on a MAP, each one corresponding to a particular place in the "REALITY" the map is a map over.

We reserve the symbol "f" to denote the mapping process or machine, while f(5) is called the function value (what pops out when putting in the number 5 in the function machine)
------------------------------------------------------------------------------------
You can therefore regard a function as a (tremendously huge!) COLLECTION of defining equalities, the principle for collecting correct equalities being the machine f.
-------------------------------
A DEFINING equality is one by which a new symbol is given a fixed meaning relative to previously established symbols.

Last edited: Oct 1, 2013
20. Oct 8, 2013

### thedy

Once again,really thanks for answers and for patient...