Function concept and equation concept

[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.
Thanks to all for answers...

 

[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

 

[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.

 

[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.

Do you follow so far?

 

[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

 

[thedy]

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

 

[Mark44]

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

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).

 

[Mark44]

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

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.

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?

No, this equation doesn't represent a function. There is only one number (5) that you can put into 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.

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.

 

[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."

 

[thedy]

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.

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...
Thanks for answers...
So in function we can have infinite options to have output,whereas in equation we have output only one,and fixed?

 

[jbriggs444]

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.

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.

So in function we can have infinite options to have output,whereas in equation we have output only one,and fixed?

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.

 

[Mark44]

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...

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.

Thanks for answers...
So in function we can have infinite options to have output,whereas in equation we have output only one,and fixed?

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.

 

[thedy]

Thanks for answers again.
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 definately 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.

 

[Mark44]

Thanks for answers again.
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?

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 do not explain better,what I mean,but it seems like every expresion we can change with name of the function.

I don't understand what you're trying to say.

Also nRT can be labeled like function by some sign...

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.

But of course,we definately 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.

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.

 

[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?

 

[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.

 

[Mark44]

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.

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.

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?

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.

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.

Your equation is equivalent to x = 5, which represents a single point on the number line.

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?

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.

 

[Mark44]

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.

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.

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?

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.

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.

Your equation is equivalent to x = 5, which represents a single point on the number line.

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?

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.

 

[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.

 

[thedy]

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

 

[arildno]

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

As a final note:
We are so used to think of "f(x)=something nice", that we tend to forget the two fundamental parts in a "function":
Namely, you've got a SET of argument values (the x's) and another set of function values (the f(3), f(5) and so on), and if we are LUCKY enough, we might deduce the principle, or rule, f, that connects together these two sets correctly.

But, these two sets, that of argument values and that of function values are EASILY seen to appear in what you can call a data series:

Suppose you buy a plant. At different times "t", you measure its height "h".

You can regard the h's as function values for the argument values t's, and can set up a nice double series of data points containing all that information.

You do NOT strictly need the explicit principle f that allows you to CALCULATE the "h" from the corresponding "t", it might not a have a very nice form at all!

But, and that you can be assured of, there IS such a rule f that maps over the t's to the h's, and that is our machine. It's just a "black box" in such a case, where we are not familiar with the interior calculating machinery, but we can read off the function values that correspond to the argument values easily enough without it. WE would be hard pressed, though, to say how high the plant will be 3 days from now WITHOUT knowledge of the machinery hidden in f..

 

[kashan123999]

10=2x...why it is not a constant function?

2nd question: if we solve for x...can x=5 be regarded as a constant function?

 

[Mark44]

10=2x...why it is not a constant function?

2nd question: if we solve for x...can x=5 be regarded as a constant function?

Please start at the beginning of this thread and read all of it. That question was asked and answered in this thread.

 

[thedy]

Hi,I am wondering what is definition of velocity and what is a function of velocity.For example:v=s/t...this is definition of velocity,but what is the difference between definition of velocity and velocity as function of something in physics?How could I clasify these concepts?Everywhere,every velocity problem is initiated by ,,suppose definition of velocity v=s/t"...and subsequently this equation is transformated to some form(I mean differential form),and later on,this form is called velocity as function of something.It seems like v=s/t is not a technique to solve velocity problems.Definition of some physical phenomena is not the same,like physical phenomena as function of something?
My question seems complicated,but v=s/t is not considered by you,physicists to be a real function?From math view of point it is a function,but the term ,,definition" mislead me.Well,what is the difference in physics between definition of something and ,,real" function of something...
Thanks for your time,patient and struggle to understand my weird question.

 

[Mark44]

The formula v = s/t gives you the average velocity. For example, if you drive a car 105 miles (s) in 1.5 hours (t), your average velocity is 70 mi/hr. In this context, average velocity and average speed are the same thing.

It's not very likely that during the trip you were doing 70 mph all the time. At some places you might have been going a little faster, and at others, going a little slower. So your instantaneous velocity at these places would have been different from your average velocity.

If s(t) represents your position at time t, then the instantaneous velocity at a time t₀is v(t₀), which is defined as
$$v(t_0) = \lim_{h \to 0} \frac{s(t_0 + h) - s(t_0)}{h}$$

In other words, velocity v(t) is the derivative of the position function s(t).

 

[thedy]

The formula v = s/t gives you the average velocity. For example, if you drive a car 105 miles (s) in 1.5 hours (t), your average velocity is 70 mi/hr. In this context, average velocity and average speed are the same thing.

It's not very likely that during the trip you were doing 70 mph all the time. At some places you might have been going a little faster, and at others, going a little slower. So your instantaneous velocity at these places would have been different from your average velocity.

If s(t) represents your position at time t, then the instantaneous velocity at a time t₀is v(t₀), which is defined as
$$v(t_0) = \lim_{h \to 0} \frac{s(t_0 + h) - s(t_0)}{h}$$

In other words, velocity v(t) is the derivative of the position function s(t).

Thanks,but I thought something else a bit.In one book,which I read is written something like this:,,expression for physical dimension do not provide formulas for actually calculating the quantities in question:you do not find a value for acceleration by dividing a length by the square of time."
Well it seems,like s/t or v/t is only dimension and not actual formula.I am confused :/
Thanks for answers again

 

[Mark44]

Thanks,but I thought something else a bit.In one book,which I read is written something like this:,,expression for physical dimension do not provide formulas for actually calculating the quantities in question:you do not find a value for acceleration by dividing a length by the square of time."
Well it seems,like s/t or v/t is only dimension and not actual formula.I am confused :/
Thanks for answers again

There are two things going on here: the value of the velocity or acceleration, and the units for these quantities.

The dimension for velocity is l/t : length divided by time.
The dimension for acceleration is l/t²: length divided by time squared.

You can find the value of the average velocity by dividing the distance traveled (final position - initial position) by the elapsed time. Similarly, you can find the average acceleration by dividing the change in velocity (v_final - v_initial) by the elapsed time. These are not the instantaneous velocity or acceleration, both of which you would calculate by taking a derivative.

 

[Stephen Tashi]

A function (from the real numbers to the real numbers) is a (perhaps infinite) set of ordered pairs of real numbers such that no two distinct pairs have the same first element.

An equation in two real variables x,y is a statement about the two real variables. The set pairs of real numbers that make the statement true is the solution set of the equation. The solution set (which can be infinite) does not have the requirement that no two of its distinct ordered pairs have the same first element. For example, the solution set of the equation $x^2 = y^2$ contains the ordered pairs (1,1), (1,-1)

The most common way to define both functions and equations is write a algebraic expression like y = mx + b. Whether y = mx + b refers to the definition of a function or the definition of an equation depends on the context where it appears. That expression by itself does not make it clear whether a function or an equation is being discussed.

Both functions and equations can be defined with various restrictions and complicated conditions. When conditions are added to an equation, the totality of the conditions is often called a "set of equations" or "an equation with constraints". However, abstractly an equation with conditions can be considered to be a single statement by considering all the conditions joined together with "AND".

You may find books that carelessly use terminology such as "graph the function $y^2 = x^2$" when the proper terminology would be "graph the relation...". or "graph the solution set of...". There are also books that use terminology like "graphing equations" when what they actually mean is "graphing functions".