Function notation and terminology
NewAtMath
I have a feeling that my questions are again pointless and that I'm just nitpicking, but I do have a problem distinguishing between what's important and what isn't
I started learning linear functions and so far the subject seems reasonably easy, but there are some confusions with the function terminology and notation:
1)Say we have equation 'y = x + 1'
a) Must we, whenever we want to deal with the above equation as a function, declare 'y = x + 1' as a function by using the following notation ( or some similar notation ):
f(x) = x + 1
b) Now, is the name of a function 'f' or 'y'? Meaning, 'y' is a name of dependable variable, while the term function suggests an action of sort, so I don't understand how we can call 'y' a function ( some sites claim that as far as notation goes, 'y' and 'f' mean the same thing ).
c) I would argue that f(x) does mean the same as 'y', but at the same time f(x) doesn't represent the function itself ( the action ), but the result of a function at given 'x' ( the result of an action). And yet my book makes no distincion ( for example when introducing some problem, it begins with: 'given a function f(x) = 2x + 1...' )
2) Say we have the following two declarations:
f:x --> y^2 and f(x) = x + 1
a) I would assume 'f' represents the function itself and 'f:x --> y^2' tells what this function does , while f( x ) just means specific value.
b) But in any case, why don't we instead of 'f:x --> y^2' write 'f = x + 1'. Because '=' operator would suggest, that some value at given 'x' is assigned to the function itself?
3) If we are given the problem 'y = 2x + 1, find ( 6, y )'
f(x) = 2x + 1 = 2 * 6 + 1 = 13
y = 13
If we were nitpicking and be kinda purists, would the result 'y = 13' be considered wrong, since I haven't declared 'y' as 'y = f(x)'?
thank you
Barb Knox
<4694867.12276553195...@nitrogen.mathforum.org>,
NewAtMath <seve...@gmail.com> wrote:
> Greetings
>
> I have a feeling that my questions are again pointless and that I'm just
> nitpicking, but I do have a problem distinguishing between what's important
> and what isn't
>
> I started learning linear functions and so far the subject seems reasonably
> easy, but there are some confusions with the function terminology and
> notation:
>
>
> 1)Say we have equation 'y = x + 1'
>
> a) Must we, whenever we want to deal with the above equation as a function,
> declare 'y = x + 1' as a function by using the following notation ( or some
> similar notation ):
>
> f(x) = x + 1
If you want to refer to the function of x whose value is x + 1 then yes,
you need some notation such as this to (A) name the function, and (B)
specify the variable(s) that stand for the argument(s) of the function.
(Note that there are ways to specify functions without naming them. For
example, "Lambda x (x + 1)" is an unnamed version of your function f.)
> b) Now, is the name of a function 'f' or 'y'? Meaning, 'y' is a name of
> dependable variable,
> while the term function suggests an action of sort,
Not really. It sounds like you know some programming, which can be a
Bad Thing when trying to learn mathematics. In mathematics, functions
are NOT actions, and "=" is NOT assignment (e.g., in mathematics "x = y"
and "y = x" are precisely equivalent).
> so I
> don't understand how we can call 'y' a function ( some sites claim that as
> far as notation goes, 'y' and 'f' mean the same thing ).
This is a common way of talking, which can be confusing for beginners,
but it is harmless when there is no ambiguity. In "y = x + 1" it is
perfectly reasonable to refer to "y as a function of x". It is also
reasonable in this case to refer to "x as a function of y" (since here x
= y - 1).
> c) I would argue that f(x) does mean the same as 'y', but at the same time
> f(x) doesn't represent the function itself ( the action ), but the result of
> a function at given 'x' ( the result of an action).
You're right that "f" is the function, whereas "f(x)" is the value of
the function for a given x. (But see my warning above about "actions").
> And yet my book makes no
> distincion ( for example when introducing some problem, it begins with:
> 'given a function f(x) = 2x + 1...' )
Here, the "given x" is actually a free variable: FOR ALL x (in the
appropriate domain, which for "x" is usually the real numbers), f of
that x equals 2x + 1. (Sadly, the distinction between variables and
constants is often left to context and convention: w, x, y, z are
usually variables; a, b, c are arbitrary constants; and pi is a
specific constant.)
> 2) Say we have the following two declarations:
>
> f:x --> y^2 and f(x) = x + 1
>
>
> a) I would assume 'f' represents the function itself and 'f:x --> y^2' tells
> what this function does , while f( x ) just means specific value.
Except, in the second one, x is a free variable, so in fact it defines f
for all x. (And again, mathematical functions don't DO, they just ARE.)
> b) But in any case, why don't we instead of 'f:x --> y^2' write 'f = x + 1'.
> Because '=' operator would suggest, that some value at given 'x' is assigned
> to the function itself?
You could say f = lambda x (x + 1), since both sides of the "=" are
functions. But "f = x + 1" is a type mismatch, since f is a function
but x + 1 is some value. And note the ambiguity if there are more than
one argument: f(x,y) = 2x + y or f(y,x) = 2x + y are unambiguous,
but f = 2x + y is ambiguous. (And remember, "=" does NOT mean
assignment; it indicates a relationship.)
3) If we are given the problem 'y = 2x + 1, find ( 6, y )'
>
> f(x) = 2x + 1 = 2 * 6 + 1 = 13
> y = 13
>
> If we were nitpicking and be kinda purists, would the result 'y = 13' be
> considered wrong, since I haven't declared 'y' as 'y = f(x)'?
By convention, the pair (..., ...) stands for (x, y). So, you are
asked to find the pair where x = 6 and y = something. To be picky, I
would say the correct answer is (6, 13), since you are asked to find a
pair, not a single value.
> thank you
HTH.
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
[Mr.] Lynn Kurtz
wrote:
>Greetings
>
>I have a feeling that my questions are again pointless and that I'm just nitpicking, but I do have a problem distinguishing between what's important and what isn't
Sincere questions are never pointless.
>I started learning linear functions and so far the subject seems reasonably easy, but there are some confusions with the function terminology and notation:
>
>
>1)Say we have equation 'y = x + 1'
No need for the ' symbols here and below.
>a) Must we, whenever we want to deal with the above equation as a function, declare 'y = x + 1' as a function by using the following notation ( or some similar notation ):
>
>f(x) = x + 1
It is certainly convenient to use this notation e.g., find f(2), f(3)
etc. as opposed to asking what is y when x = 2 etc.
>b) Now, is the name of a function 'f' or 'y'? Meaning, 'y' is a name of
>dependable variable,
that's "dependent". The name of the function is f.
> while the term function
>suggests an action of sort, so I don't understand how we can call 'y' a function ( some sites claim that as far as >notation goes, 'y' and 'f' mean the same thing ).
>
They should say that y and f(x) are the same thing.
>c) I would argue that f(x) does mean the same as 'y', but at the same time f(x) doesn't represent the function itself ( the action ), but the result of a function at given 'x' ( the result of an action). And yet my book makes no distincion ( for example when introducing some problem, it begins with: 'given a function f(x) = 2x + 1...' )
>
And you would be correct. When you use the notation y = f(x) = 2x + 1,
the name of the function is " f ". Some algebra books are indeed
sloppy about that and gloss over the difference.
>2) Say we have the following two declarations:
>
>f:x --> y^2 and f(x) = x + 1
you likely mean f: x --> x^2
>
>a) I would assume 'f' represents the function itself and 'f:x --> y^2' tells what this function does , while f( x ) just means specific value.
Yes, assuming you mean x^2, f is the name of the function, x --> x^2
describes it, and f(x) represents its value at x.
>
>b) But in any case, why don't we instead of 'f:x --> y^2' write 'f = x + 1'. Because '=' operator would suggest, that some value at given 'x' is assigned to the function itself?
>
No, we don't write f = x + 1 because a function f is not the same
thing as its value at a particular x. You would write f(x) = x + 1.
>3) If we are given the problem 'y = 2x + 1, find ( 6, y )'
That is an odd way to phrase the question if that's what your book
does.
>f(x) = 2x + 1 = 2 * 6 + 1 = 13
The second = above is incorrectly used. You need to write two
statements:
f(x) = 2x + 1
f(2) = 2*6 + 1 = 13.
>y = 13
>
>If we were nitpicking and be kinda purists, would the result 'y = 13' be considered wrong, since I haven't declared 'y' as 'y = f(x)'?
Given the way the problem is stated, it would be hard to make that
nit-pick.
--Lynn
Frederick Williams
> (Note that there are ways to specify functions without naming them. For
> example, "Lambda x (x + 1)" is an unnamed version of your function f.)
As is x |-> x + 1 which the OP may see more often than he sees
lambda.
--
But you see, I can believe a thing without understanding it.
It's all a matter of training.
--Lord Peter Wimsey in Dorothy L Sayers' _Have His Carcase_
Frederick Williams
>
> On Tue, 25 Nov 2008 18:21:29 EST, NewAtMath <seve...@gmail.com>
> wrote:
>
> >2) Say we have the following two declarations:
> >
> >f:x --> y^2 and f(x) = x + 1
>
> you likely mean f: x --> x^2
Some write f: x |-> x^2 and reserve --> for domain --> codmain.
(I'm writing that for the benefit of the OP, not because I think you
don't know it.)
Frederick Williams
> 1)Say we have equation 'y = x + 1'
>
> a) Must we, whenever we want to deal with the above equation as a function, declare 'y = x + 1' as a function by using the following notation ( or some similar notation ):
>
> f(x) = x + 1
It depends what you are doing with the equation. Note that you can get
two functions out of it, the one you mention and its inverse
g(y) = y - 1,
with y = x^2 you can't.
> yet my book makes no distincion ( for example when introducing some problem, it begins with: 'given a function f(x) = 2x + 1...' )
This is just short for 'given a function f such that f(x) = 2x + 1...'.
It's harmless when you get used to it. One would also like to see f's
domain and codomain specified, but the context may do that and, again,
one gets used to it.
[Mr.] Lynn Kurtz
<ku...@asu.edu.invalid> wrote:
Two days later I notice a typo:
>f(x) = 2x + 1
>f(2) = 2*6 + 1 = 13.
should be f(6) = 2*6 + 1 = 13
--Lynn
NewAtMath
I really appreciate all your help. May just ask the following and then wrap it up?
>>2) Say we have the following two declarations:
>>
>>f:x --> y^2 and f(x) = x + 1
>
>you likely mean f: x --> x^2
Yes
>>3) If we are given the problem 'y = 2x + 1,
>>find ( 6, y )'
>
>That is an odd way to phrase the question if that's what
>your book does.
I made up a problem ( English is not my first language so … I know that’s not an excuse :( )
>>f(x) = 2x + 1 = 2 * 6 + 1 = 13
>
>The second = above is incorrectly used. You need to
>write two statements:
>
>f(x) = 2x + 1
>f(2) = 2*6 + 1 = 13.
>
>>y = 13
>>
>>If we were nitpicking and be kinda purists, would the
>>result 'y = 13' be considered wrong,
>>since I haven't declared 'y' as 'y = f(x)'?
>
>Given the way the problem is stated, it would be hard to
>make that nit-pick.
Could you elaborate on what you mean by that?
In any case, if problem was more correctly stated, would my nitpicking be correct?
>>1)Say we have equation 'y = x + 1'
>>
>>a) Must we, whenever we want to deal with the above
>>equation as a function, declare 'y = x + 1' as a
>>function by using the following notation ( or some
>>similar notation ):
>>
>>f(x) = x + 1
>
>If you want to refer to the function of x whose value is
>x + 1 then yes, you need some notation such as this to
>(A) name the function, and (B) specify the variable(s)
>that stand for the argument(s) of the function.
So, if we don't use any special function notation, but just use expression "y = x + 1", then could we still claim that "y = x + 1" represents some unnamed function, or must we at least use "Lambda x (x + 1)" to be able to make such claim?
>function of x
Is there any special reason why we say "function f OF X" and not just, for example, "function f"?
>>b) Now, is the name of a function 'f' or 'y'? Meaning,
>>'y' is a name of dependable variable, while the term
>>function suggests an action of sort,
>
>Not really. It sounds like you know some programming,
>which can be a Bad Thing when trying to learn
>mathematics. In mathematics, functions are NOT
>actions,
That's a bit confusing. Doesn't the very definition of a name suggests an action of sort? I imagined it as some 'entity, machine if you will', which when given certain input (x), it produces certain output/product ( y ). What does the name function then stand for in this context?
>> And yet my book makes no
>> distincion ( for example when introducing some
>>problem, it begins with:
>> 'given a function f(x) = 2x + 1...' )
>
>Here, the "given x" is actually a free variable: FOR ALL
>x (in the appropriate domain, which for "x" is usually
>the real numbers), f of that x equals 2x + 1.
>>2) Say we have the following two declarations:
>>
>> f:x --> y^2 and f(x) = x + 1
>>
>>
>> a) I would assume 'f' represents the function itself
>>and 'f:x --> y^2' tells what this function does , while
>>f( x ) just means specific value.
>
>Except, in the second one, x is a free variable, so in
>fact it defines f for all x. (And again, mathematical
>functions don't DO, they just ARE.)
By second one you mean "f(x) = x + 1"? So "f(x) = x + 1" INDIRECTLY defines/declares "f", while "f:x --> x^2" defines/declares it directly?
>>so I
>>don't understand how we can call 'y' a function ( some
>>sites claim that as far as notation goes, 'y' and 'f'
>>mean the same thing )
>
>This is a common way of talking, which can be confusing
>for beginners, but it is harmless when there is no
>ambiguity. In "y = x + 1" it is perfectly reasonable to
>refer to "y as a function of x". It is also
>reasonable in this case to refer to "x as a function of
>y" (since here x = y - 1).
But even though it is understood by everybody what we really mean to say by "y as a function of x" and is thus in a way correct, it is still ( in the strictest sense ) wrong?
>And remember, "=" does NOT mean assignment; it
>indicates a relationship.
Are you referring to its meaning with functions, or in general? Thus, if we have equation 'a = 20', does here '=' also represent equality?
>>yet my book makes no distincion ( for example when
>>introducing some problem, it >>begins with: 'given a
>>function f(x) = 2x + 1...' )
>
>This is just short for 'given a function f such that
>f(x) = 2x + 1...'. It's harmless when you get used to >it.
So basically "f(x) = x + 1" INDIRECTLY defines/declares "f"?
thank you
[Mr.] Lynn Kurtz
wrote:
>>>3) If we are given the problem 'y = 2x + 1,
>>>find ( 6, y )'
>>
>>That is an odd way to phrase the question if that's what
>>your book does.
>
>I made up a problem ( English is not my first language so ? I know that?s not an excuse :( )
>
Your English is much better than my effort at any foreign language
would be.
>>>If we were nitpicking and be kinda purists, would the
>>>result 'y = 13' be considered wrong,
>>>since I haven't declared 'y' as 'y = f(x)'?
>>
>>Given the way the problem is stated, it would be hard to
>>make that nit-pick.
>
>Could you elaborate on what you mean by that?
By asking for (6, y), you have implicitly indicated that y = f(x)
although you didn't declare it explicitly.
>In any case, if problem was more correctly stated, would my nitpicking be correct?
Yes, I suppose so. But the difficulty arises because the problem is
poorly stated. You should have phrased it either as:
"Find y when x = 6" or you should have stated f(x) = 2x + 1 and asked
to find f(6).
--Lynn
Frederick Williams
>
> I made up a problem ( English is not my first language so … I know that’s not an excuse :( )
Oh, it is! In any case, never mind because you write better English
than some posters whose first language is English.
> Is there any special reason why we say "function f OF X" and not just, for example, "function f"?
There may be. One might want to say "function of a real variable" in
case the domain is in doubt.
> So basically "f(x) = x + 1" INDIRECTLY defines/declares "f"?
I don't know what this notion of INDIRECTLY means.
f(x) = x + 1
and
f: x |-> x + 1
convey the same thing to me. (Don't say "declare" say "define".)
Barb Knox
<18225719.1227741811...@nitrogen.mathforum.org>,
NewAtMath <seve...@gmail.com> wrote:
[snip]
> >>1)Say we have equation 'y = x + 1'
> >>
> >>a) Must we, whenever we want to deal with the above
> >>equation as a function, declare 'y = x + 1' as a
> >>function by using the following notation ( or some
> >>similar notation ):
> >>
> >>f(x) = x + 1
> >
> >If you want to refer to the function of x whose value is
> >x + 1 then yes, you need some notation such as this to
> >(A) name the function, and (B) specify the variable(s)
> >that stand for the argument(s) of the function.
>
>
> So, if we don't use any special function notation, but just use expression "y
> = x + 1", then could we still claim that "y = x + 1" represents some unnamed
> function, or must we at least use "Lambda x (x + 1)" to be able to make such
> claim?
'y = x + 1' (or equivalently, 'x + 1 = y') does not represent a single
function. It represents a relationship between x and y, which happens
to be a functional one in either direction:
f(x) = x + 1
g(y) = y - 1
For a function of 1 argument, you need to specify both (A) the variable
which stands for that argument, and (B) the value of the function in
terms of that variable. Both the 'f(var) = expression' and
'lambda var expression' forms do just that.
> >function of x
>
> Is there any special reason why we say "function f OF X" and not just, for
> example, "function f"?
Yes, because it says that wherever 'x' appears in the function
definition it stands for the argument of the function.
But the phrase "the function f" is perfectly fine, and is appropriate
when talking about 'f' itself rather than a particular definition of it.
> >>b) Now, is the name of a function 'f' or 'y'? Meaning,
> >>'y' is a name of dependable variable, while the term
> >>function suggests an action of sort,
> >
> >Not really. It sounds like you know some programming,
> >which can be a Bad Thing when trying to learn
> >mathematics. In mathematics, functions are NOT
> >actions,
>
> That's a bit confusing. Doesn't the very definition of a name suggests an
> action of sort?
No. You can think of 'f' as a set of pairs of numbers, with the first
number in each pair being an 'x' value and the second one being the
corresponding 'y'. All the x-y pairs are "already there" in the set, so
no action (e.g. computation) is required.
> I imagined it as some 'entity, machine if you will', which
> when given certain input (x), it produces certain output/product ( y ). What
> does the name function then stand for in this context?
In some mathematical contexts (NOT including introductory algebra),
"function" does indeed mean "computable function". But for your present
purposes, it just means a particular kind of relation, which you can
consider to be "already there".
> >> And yet my book makes no
> >> distincion ( for example when introducing some
> >>problem, it begins with:
> >> 'given a function f(x) = 2x + 1...' )
> >
> >Here, the "given x" is actually a free variable: FOR ALL
> >x (in the appropriate domain, which for "x" is usually
> >the real numbers), f of that x equals 2x + 1.
>
>
> >>2) Say we have the following two declarations:
> >>
> >> f:x --> y^2 and f(x) = x + 1
> >>
> >>
> >> a) I would assume 'f' represents the function itself
> >>and 'f:x --> y^2' tells what this function does , while
> >>f( x ) just means specific value.
> >
> >Except, in the second one, x is a free variable, so in
> >fact it defines f for all x. (And again, mathematical
> >functions don't DO, they just ARE.)
>
> By second one you mean "f(x) = x + 1"?
Yes.
> So "f(x) = x + 1" INDIRECTLY
> defines/declares "f", while "f:x --> x^2" defines/declares it directly?
No, they both directly define the function relation by specifying the
value of the function for every possible value of its argument.
> >>so I
> >>don't understand how we can call 'y' a function ( some
> >>sites claim that as far as notation goes, 'y' and 'f'
> >>mean the same thing )
> >
> >This is a common way of talking, which can be confusing
> >for beginners, but it is harmless when there is no
> >ambiguity. In "y = x + 1" it is perfectly reasonable to
> >refer to "y as a function of x". It is also
> >reasonable in this case to refer to "x as a function of
> >y" (since here x = y - 1).
>
> But even though it is understood by everybody what we really mean to say by
> "y as a function of x" and is thus in a way correct, it is still ( in the
> strictest sense ) wrong?
It's not wrong, it's just another way of speaking.
> >And remember, "=" does NOT mean assignment; it
> >indicates a relationship.
>
> Are you referring to its meaning with functions, or in general?
In general; always. It's crucial to not confuse equality with
assignment.
> Thus, if we
> have equation 'a = 20', does here '=' also represent equality?
Yes indeed. This is precisely equivalent to '20 = a' (although by
convention it would very seldom be written that way).
> >>yet my book makes no distincion ( for example when
> >>introducing some problem, it >>begins with: 'given a
> >>function f(x) = 2x + 1...' )
> >
> >This is just short for 'given a function f such that
> >f(x) = 2x + 1...'. It's harmless when you get used to it.
>
>
> So basically "f(x) = x + 1" INDIRECTLY defines/declares "f"?
No, there's nothing indirect about it. It fully defines the functional
relation 'f' on the real numbers.
Frederick Williams
I've reformatted your post because it's lines are too long for my
newsreader.
> That's a bit confusing. Doesn't the very definition of a name
> suggests an action of sort?
Perhaps. You may have been given a poor definition. In any case that
is a matter of psychology, not mathematics.
> I imagined it as some 'entity, machine
> if you will', which when given certain input (x), it produces
> certain output/product ( y ).
Sometimes a phrase like "black box" is used, but that's just a metaphor
and you shouldn't think of machines. If a metaphor makes you think of
them, it's a bad metaphor. One can prove that there are functions not
computable by a machine. See any book o recursion theory.
> What does the name function then
> stand for in this context?
It would name the "entity". If there is more than one you'll need to
call them something to distinguish between them. If all functions were
associated with formulae in some unambiguous way then I suppose the
formula might be used as a name for the function (*) but there are more
functions than formulae.
(* But it's not a good idea. Consider the formula
x + y
is that the function of x that adds y to it, the function of y that adds
x to it or the function of x and y that adds them? Lambda notation (and
others) fixes these problems.)
NewAtMath
>>>>yet my book makes no distincion ( for example
>>>>when introducing some problem, it >>begins with:
>>>>'given a function f(x) = 2x + 1...' )
>>>
>>>This is just short for 'given a function f such that
>>>f(x) = 2x + 1...'. It's harmless when you get used
>>>to it.
>>
>>
>>So basically "f(x) = x + 1" INDIRECTLY
>>defines/declares "f"?
>
>No, there's nothing indirect about it. It fully
>defines the functional relation 'f' on the
>real numbers.
>
>>So "f(x) = x + 1" INDIRECTLY
>>defines/declares "f", while "f:x --> x^2"
>>defines/declares it directly?
>
>No, they both directly define the function relation
>by specifying the value of the function for every
>possible value of its argument.
>
>>So basically "f(x) = x + 1" INDIRECTLY
>>defines/declares "f"?
>
>I don't know what this notion of INDIRECTLY means.
>
An analogy would be someone saying she's lived in London since her birth. This indirectly tells us that she is British.
Thank you all for helping me out with this
cheers