Functions which relate to calculus: Questions about Notation

Hi. I'm self-studying functions which relate to calculus. Let me post what I feel I know and what I'm not grasping yet. Please correct any mistakes I'm making.

I'm just talking real numbers: A function is a rule that takes an input number and sends it to another number. We can describe it perfectly well with words. Often the input number can be denoted by the letter x. When a graph is drawn of the output number that is often denoted by the letter y.

Take the rule/function "square and add two". In mathematical notation that function can be written as:

x² + 2.

Okay, well, I get that. Now, it is my understanding that you can denote or refer (right words?) a function by a single character, such as "f". Like in the following sentence - "If we let x run through a function f,...." Here "f" does not stand for, or and is not the equivalent of the precise rule, it's only refering to a function in general.

Now we get to a notation that I'm not clear about:

f (x) = x² + 2

F (x) does define the rule. But, what gets me, is that this looks like an algebraic equation, something where with manipulation skills you could isolate x. But, for that to be true, you would need to see the left hand side as f times x. I guess I'm not really grasping what f (x) is. I do know, I think, that it is equal to x² + 2. In some kind of way, but not in an algebraic way (I've concluded).

I'm concluding f = x² + 2 is incorrect for defining the function, but:

f (x) = x² + 2 is correct. For some reason I'm not grasping. What is (x) doing?

Thanks for any assistance. Rich

If ##f(x) = 2x +4##
Then ##f(3) = 2\cdot 3 +4 = 10##
and ##f(2a+1) = 2 (2a+1) +4 = 4a +6##
##f(f(x))= 2f(x) +4 = 2(2x+4)+4=4x+12##
And so on

It is not f times x.

richard9678 said:

Hi. I'm self-studying functions which relate to calculus. Let me post what I feel I know and what I'm not grasping yet. Please correct any mistakes I'm making.

I'm just talking real numbers: A function is a rule that takes an input number and sends it to another number. We can describe it perfectly well with words. Often the input number can be denoted by the letter x. When a graph is drawn of the output number that is often denoted by the letter y.

Take the rule/function "square and add two". In mathematical notation that function can be written as:

x² + 2.

Okay, well, I get that. Now, it is my understanding that you can denote or refer (right words?) a function by a single character, such as "f". Like in the following sentence - "If we let x run through a function f,...." Here "f" does not stand for, or and is not the equivalent of the precise rule, it's only refering to a function in general.

Now we get to a notation that I'm not clear about:

f (x) = x² + 2

F (x) does define the rule. But, what gets me, is that this looks like an algebraic equation, something where with manipulation skills you could isolate x. But, for that to be true, you would need to see the left hand side as f times x. I guess I'm not really grasping what f (x) is. I do know, I think, that it is equal to x² + 2. In some kind of way, but not in an algebraic way (I've concluded).

I'm concluding f = x² + 2 is incorrect for defining the function, but:

f (x) = x² + 2 is correct. For some reason I'm not grasping. What is (x) doing?

Thanks for any assistance. Rich

##f(x)## is the standard mathematical notation to represent the function value of ##f## acting on ##x##. It could be written in other ways, but that's the standard.

OK, what I get is this: f (x) = x2 + 2 is merely defining the rule only. Specifically - as it stands - it's not capable of being used to work out the output number. In order to work out the output number, you must substitute (x) for an actual input number. f (x) is not to be seen as f times x, but merely as a kind of syntax. I think I've now got it.

Someone asks what is the rule for function f, or the definition of f? You write f (x) = x^2 + 2

Someone asks what is the output number (say "y") when x is 3. You write f (3) = 3^2 + 2 = 11 = y. f (3) is not to read as f times 3.

richard9678 said:

##f(x) = x^2 + 2## is correct. For some reason I'm not grasping. What is (x) doing?

You already answered it:

richard9678 said:

A function

##f##

richard9678 said:

is a rule that takes an input number

##x##

richard9678 said:

and sends it

##=##

richard9678 said:

to another number.

##x^2+2.##

$$
\underbrace{f}_{function} \;\underbrace{(x)}_{takes\;x} \underbrace{=}_{sends\;it\;to} \underbrace{x^2+2}_{another \;number}
$$
The important aspect, however, is that a function cannot attach two different numbers to the same value of ##x.## This might sound obvious now but will become important if ##x## represents e.g. classes of integers like for instance remainders of a division rather than a single number.

Example: ##f(x)=\sqrt{x}=+\sqrt{x}## is a function. ##r(x)=\pm\sqrt{x}## is no function.

Thanks for everyone's input. I can also see that f(x) = x^2 + 2 is just a variation on "The function x → x^2 + 2". Which of course we use when defining the function. And that we can focus just on f (x), as in, evaluate f(3) (in light of the precise rule). And that we can engage in analysis of what values of x (the domain) are allowed given a particular rule. In other words, x is very much the subject in f(x). The notation allows discussion and analysis of x. I have gained some key insights in this thread I think. Thanks.

richard9678 said:

Thanks for everyone's input. I can also see that f(x) = x^2 + 2 is just a variation on "The function x → x^2 + 2". Which of course we use when defining the function. And that we can focus just on f (x), as in, evaluate f(3) (in light of the precise rule). And that we can engage in analysis of what values of x (the domain) are allowed given a particular rule. In other words, x is very much the subject in f(x). The notation allows discussion and analysis of x. I have gained some key insights in this thread I think. Thanks.

There are actually quite a few ways to describe a function. Let's stick with your example ##f(x)=x^2+2.## Then

• ##f## is a function.
• ##f(x)## is a function. That is most common but actually a bit sloppy. It is not the function but the function value of ##x.## Since ##x## isn't further specified, people call it function, too.
• ##y=x^2+2## is a function.
• ##y(x)=x^2+2## is a function.
• ##x \longmapsto x^2+2 ## is a function.
• ##f\, : \,x \longmapsto x^2+2 ## is a function.
• And here comes the real definition:
  $$
  f=\{(x,x^2+2)\in \mathbb{R}\times \mathbb{R}\,|\,x\in \mathbb{R}\}
  $$

All these versions are used. The most complicated one, the last one, is the set-theoretical definition, a subset of ##\mathbb{R}\times \mathbb{R}=\mathbb{R}^2,## the two-dimensional plane. That's why we can draw ##f(x)=x^2+2## in a coordinate system. A function is a specific relation. One that has only one value ##f(x)## for one ##x##.

In f(x) = x^2+2, every symbol is a number, except f, which stands for the word "function". That's what I'm concluding. So in it's essence x = x^2+2, or x→x^2+2. But you should not write x= x^2+2, you need to write f(x) = x^2+2.

richard9678 said:

Take the rule/function "square and add two". In mathematical notation that function can be written as:

x2 + 2.

Just so you are absolutely clear, x**2 + 2 is NOT a function, it is an expression. Generally speaking the syntax is

function declaration = expression

so, for example f(x) = x**2 + 2.

richard9678 said:

In f(x) = x^2+2, every symbol is a number, except f, which stands for the word "function". That's what I'm concluding. So in it's essence x = x^2+2, or x→x^2+2. But you should not write x= x^2+2, you need to write f(x) = x^2+2.

One could also write g(x)=x^2+2. f and g are just names.
x=x^2+2 is an algebraic equation, not a function.
You need to recognize all the forms, because you will see them all.

richard9678 said:

In f(x) = x^2+2, every symbol is a number, except f, which stands for the word "function". That's what I'm concluding. So in it's essence x = x^2+2, or x→x^2+2. But you should not write x= x^2+2, you need to write f(x) = x^2+2.

Yes.

One cannot write ##x=x^2+2.## That would be an equation, not a function.

A function is an assignment, so ##x\longmapsto x^2+2## shows this mapping. What I wrote was ##f=\{(x\, , \,x^2+2)\,|\,x\in \mathbb{R}\}## which is a set of pairs. A function is such a set of pairs: every ##x## corresponds to ##x^2+2## so we can write them as pairs of numbers: ##\{(0,2)\, , \,(1,3)\, , \,(2,6)\, , \,(-2,6)\, , \,(-3,11)\, , \,\ldots\}## Of course, there are incredibly many pairs in this set, nevertheless, they represent the function ##f(x)=x^2+2.## It is just another perspective.

The issue you may be running into is what I call the dummy variable issue. It especially crops up in infinitesimal based treatments of calculus. You are correct in your formal definition of a function f(x) as a rule that given any x then another number f can be assigned to each x. But say x is a function of y ie x(y). Then f(x) = f(x(y)). f now is still a function of x but since x is a function of y, f can be considered a function of y ie f(y).

This can be confusing sometimes. A text may simply write f, instead of f(x) or f(y) leaving up in the air exactly what it depends on. Without going into details in calculus texts you may see something like df. No variable is specified. If f depends on x ie f(x), df = f(x + dx) - f(x). If f depends on y, df = f(y + dy) - f(y). It is left up in the air what it depends on. It can confuse beginners, but once you get used to it it is quite useful. Differential equations are an important topic in calculus and mathematics in general. You probably have not come across them yet - but if you continue along the path of learning calculus you undoubtably will. There is a technique of solving differential equations called separation of variables where this notation of leaving up in the air exactly what a function depends on comes into its own. For now simply take f(x) as a rule that to each value of x a value is assigned to f(x), but as you progress things will get more sophisticated.

As one mathematician expressed it in advice to students: “Please forget everything you have learnt in school; for you have not learnt it.”– Edmund Landau in Grundlagen der Analysis (Foundations of Analysis).

In one sense depressing, but in another sense very exciting - you are always progressing and leaning. Old ideas give way to new ones, which give way to even newer ones - it is never ending.

Thanks
Bill

Hi. I'm still with this. I've done something and now cannot see the math that appears in some posts. Something to do with Mathjax. I altered something.

richard9678 said:

Hi. I'm still with this. I've done something and now cannot see the math that appears in some posts. Something to do with Mathjax. I altered something.

???

This is my latest analysis:

f(x)=x^2 declares or defines the rule.

f can mean "function". However it can mean a mathematical statement or expression such as f=x^2.

f is a fixed rule, x is a variable and the output y, is a variable.

f(x) is "f of x". However it could be more convenient to say "Of x: square" (Or multiply by 3, divide by 2 - whatever).

f(x) is not f times x. What you do with x is rule dependent. An example of "divide x by 2":

For f(x)=x/2

1) Stop writing f, it seems to confuse you. Any name will do, so choose g for simplicity.
2) g(x) is read as “g of x”
3) frequently g(x) is written as g.
4) when dealing with the plane one frequently chooses the name of the function to be y or y(x)
5) two functions that do the same thing are identical. For example g(x)=2x, g=2x, y(x)=2x and y=2x are identical. There is no point in distinguishing them other than for naming purposes.
6) Names are chosen for convenience.
For example, supercalifragilisticexpialidocius(x) is an acceptable function name. Most of the time it is not a convenient name.

richard9678 said:

However it can mean a mathematical statement or expression such as f=x^2.

This would not normally be written. f (or g or h or P or whatever) is the name of the function; f(x) is the function value with input x; the ##x^2## part is how the function value is obtained from an input value x.

richard9678 said:

However it could be more convenient to say "Of x: square"

??? The notation sometimes used is ##f : x \rightarrow x^2##.

richard9678 said:

For f(x)=x/2

x​	f(x)​	y​	f​
1​	1/2​	1/2​	x/2​
2​	2/2​	2/2​	x/2​
3​	3/2​	3/2​	x/3​

The 3rd and 4th columns are completely unnecessary, assuming that the line above the table says ##y = f(x) = \frac x 2##.

Frabjous said:

3) frequently g(x) is written as g.

Only if we want to refer to the function itself rather than the resulting value of the function.

Frabjous said:

5) two functions that do the same thing are identical. For example g(x)=2x, g=2x, y(x)=2x and y=2x are identical. There is no point in distinguishing them other than for naming purposes.

You wouldn't normally see g = 2x for the reason I gave above. The OP is likely to be confused by this mixing of the name of a function (e.g. f) with its output (e.g. f(x)).

Hi. I think I may have realised something which brings a missing clarity.

Say the function rule is "square then add two". Before the notation f(x), I imagine that the rule was spelled out and the associated mathematics was y=x^2+2. You can perceive the rule in that algebraic equation on the right hand side, (by noting the exponent and the +2). But, as I say the rule is probably spelled out.

But it's realised that some other notation is needed. Because in terms of a mathematical expression, y is the symbol for the output number, not the rule. Of course, we write something similar when we write f(x)=x^2+2. f(x)=y. But, this notation is more convenient because it does (in a mathematical way) spell out the rule. We say "f of x", which is also saying "Of x: square, then add two". The symbol, y does not say that. Although f(x) is part of an equation, f(x) does not mean f times x. We apply whatever the rule says to x. So, if we write f(x)=x/2, that is precisely what we do. We are saying "Of x:divide by two", and it results in a number value.

So, we have a situation where y = f(x) = x^2+2. All are numbers.

The notation also allows us to evaluate y by simply placing a number of our choice for x as in: evaluate f(3). y then is 3^2+2= 11.

The notation also allows us to name the rule, which y does not do. It's common the give a function / rule the name f, g, h, or whatever we want.

I think that's about it. I think I have sufficient clarity now about f(x) notation. I realise that not all equations can amount to a function. And that with some functions the domain is limited or restricted.

I also feel to say that f(x) unevaluated is a fixed or constant rule. When evaluated a number is created which is a variable. Well, x is a variable.

richard9678 said:

I also feel to say that f(x) unevaluated is a fixed or constant rule.

Not sure what you mean by that. What is the difference between a fixed rule and a rule? What is a constant rule? I doubt you mean what the term normally means in math (In calculus, the constant rule is "the derivative of any constant is zero")

richard9678 said:

Hi. I think I may have realised something which brings a missing clarity.

Say the function rule is "square then add two". ...

Well, most people just use it. They use one or even several of the notations I mentioned earlier and work with functions without thinking too much about it. If you, however, insist on a rigorous understanding then you need to understand the set-theoretical definition.

Say we have a function ##f\, : \,A\longrightarrow B## from a set ##A## to a set ##B##. Then we first consider the set of all pairs ##(a,b) \in A\times B,## the Cartesian product of ##A## and ##B.##

Definition: A function ##f\, : \,A\longrightarrow B## is a subset ##f\subseteq A\times B## such that
$$
(a,b) \in f\; \wedge \;(a,c)\in f \;\Longrightarrow b=c
$$
Other subsets that do not necessarily fulfill this condition are called relations.

This is the formal definition of a function. No rules, just pairs. The set ##\{a\in A\,|\,(a,b)\in f\text{ for some }b\in B\}## is called the domain of ##f,## the set ##B## is called the codomain, the set ##\{b\in B\,|\,(a,b)\in f\text{ for some }a\in A\}## is called the range or image of ##f.##

I still wouldn't bother with those definitions a lot and just work with functions, but if you really want to understand what a function is, then you should study their definitions instead of trying to guess a personal description hoping it is correct. The important part is indeed that specific condition that distinguishes a function from any subset of ##A\times B,## relations. It means that a function is a relation, but a relation is in general no function.

phinds said:

Not sure what you mean by that. What is the difference between a fixed rule and a rule? What is a constant rule? I doubt you mean what the term normally means in math (In calculus, the constant rule is "the derivative of any constant is zero")

I think I should have said: I also feel to say that f is a fixed or constant rule, whereas f(x) is a variable result or calculation.

All noted. In particular that g is not the same thing as g(x). I think this has been a problem for me. Also, because I know a little algebra I've been slow to see and have been somewhat confused as to what f(x) means. Yes f (or g or whatever) is the relation that amounts to a function, but f(x) is supposed to be seen as one thing, entity, whatever. (If f(x) meant f times x, then you could separate them). In other words it's ok to talk about f - it's the function you are referencing. In doing that you are not referencing f(x), the output y. That's my current grasping. f(x)=y. Does f=y? I think: No. Unless there are contextual issues when referencing f. This would mean f=x^2+2 is invalid. f(x)=x^2+2 is valid. I think like saying "By jove I think I've got it". But I'm half expecting someone to throw a spanner into the works. :-)

richard9678 said:

I also feel to say that f is a fixed or constant rule, whereas f(x) is a variable result or calculation.

Better: f is the name of the function. You don't add clarity by the use of "fixed or constant rule." It's likely that we use 'f' for historical reasons, as it's the first letter of the the word "function" (English) or "fonction" (French). If additional names are needed for other functions, subsequent letters g and h are frequently used.

richard9678 said:

All noted. In particular that g is not the same thing as g(x). I think this has been a problem for me. Also, because I know a little algebra I've been slow to see and have been somewhat confused as to what f(x) means. Yes f (or g or whatever) is the relation that amounts to a function, but f(x) is supposed to be seen as one thing, entity, whatever. (If f(x) meant f times x, then you could separate them). In other words it's ok to talk about f - it's the function you are referencing. In doing that you are not referencing f(x), the output y. That's my current grasping. f(x)=y. Does f=y? I think: No. Unless there are contextual issues when referencing f. This would mean f=x^2+2 is invalid. f(x)=x^2+2 is valid. I think like saying "By jove I think I've got it". But I'm half expecting someone to throw a spanner into the works. :-)

In order:
Yes, g and g(x) mean different things.

Yes, f and g are relations (a relation is just a mapping between two sets), but are in fact functions, which is more restrictive. For a function, each element of the domain (each input value) is paired with exactly one element in the range/codomain).

It's reasonable to be confused by parenthesized expressions at first, as f(x) and 2(xy) mean entirely different things. By now I'm sure you realize that the first is a function value while the second means 2 times the product of x and y. Context is usually helpful to distinguish whether we're talking about the argument of a function versus a multiplication.

Using your example of ##f(x) = x^2 + 2## -- the graph of this function is identical to the graph of ##y = x^2 + 2##. That is, the graph is a parabola that opens up, whose vertex is at the point (0, 2).

In this case, both f(x) and y represent ordinates (i.e., y values) on the graph. It's something of an abuse of notation to write ##y(x) = x^2 + 2##, but some textbooks might use this.

Mark44 said:

(a relation is just a mapping between two sets)

This is not true. A mapping suggests a direction. A relation is a subset of the Cartesian product of two sets. There is nothing that maps. A is a sister of B is a relation, no mapping.

Correction noted.

richard9678 said:

That's my current grasping. f(x)=y. Does f=y? I think: No. Unless there are contextual issues when referencing f. This would mean f=x^2+2 is invalid. f(x)=x^2+2 is valid. I think like saying "By jove I think I've got it". But I'm half expecting someone to throw a spanner into the works. :-)

If your conclusion is that mathematical notation is often ambiguous, you are correct, but to say f=x^2+2 is "invalid" suggests there are stringent standards for notation. It's better to say that f=x^2+2 is "bad" notation. By the standards of formal mathematical logic, the commonly encountered discussions of mathematics in textbooks are imprecise. Yes, we do have to understand the context of symbolic expressions in order to interpret them. Expressions using "=" sometimes define functions and sometimes define equations where the task is to solve for values of variables that make the claim of equality true. We can also mention the context of some computer programming languages where an expression like x=x+1 denotes an instruction. If interpreted as an equation x=x+1 would have a null solution set.

I think clarity comes from experience, which one does not have as a beginner. So, having a sense of when something is bad form helps.

The following is OK:
(1) "The function x^2+2 is..."

(2) "The function f is....."

(3) "Function f is defined by the rule f(x)=x^2+2.."

(4) "The function f (f(x)) is defined by f(x)=x^2+2."

Here (f(x)), (f(x) is within parenthesis, and is is simply saying f is in terms of identification, the same function as f(x). It does not say mathematically equal to.

The following is bad form:
(5) f=f(x) is bad form because we have used the sign of mathematical equality. Just - bad form that's all.

richard9678 said:

(1) "The function x^2+2 is..."

that is an EXPRESSION, not a function.

phinds said:

Just so you are absolutely clear, x**2 + 2 is NOT a function, it is an expression.

Stephen Tashi said:

We can also mention the context of some computer programming languages where an expression like x=x+1 denotes an instruction. If interpreted as an equation x=x+1 would have a null solution set.

More specifically, x = x + 1 would be an assignment statement. Most programming languages distinguish between the assignment operator and the equality operator. For example, C and most languages derived from C use '=' exclusively for assignment and '==' for comparison of equality.

richard9678 said:

The following is bad form:
(5) f=f(x) is bad form because we have used the sign of mathematical equality.

Yes, and no.

Technically, ##f## names a function and ##f(x)## a function value, namely the one at point ##x.## But as ##x## is considered to be arbitrary, ##f(x)## is also used to mean all possible function values, namely ##\{f(x)\},## which finally is a shortcut for ##\{(x,f(x))\}## which is another notation for the function ##f.##

##f=f(x)## is also used to mean ##x\mapsto f(x).## Blame it on typewriters. Have you ever seen what effort it takes to change a selectric typewriter? You bet that you would type ##f=f(x)## after the fifth time you changed it in order only to type ##x \mapsto f(x)## on the same page! Such things remain alive long after the real reasons for it have vanished!

Hence, people write ##f=f(x)## simply to say: ##f## is a function depending on ##x##. E.g. if we consider a bunch of parabolas ##x\longmapsto a\cdot x^2 ## parameterized by a real number ##a,## and we want to calculate their extreme values, then we need to know whether we are talking about functions ##x \longmapsto a\cdot x^2## or ##a \longmapsto a\cdot x^2.## In order to eliminate confusion, you can write ##f=f(x)=a\cdot x^2.## This is done to emphasize that the variable is ##x## and not ##a.##

I don't think you do yourself a favor with this discussion. It confuses you more than it teaches you something. Notations and language are always merely tools to express something and different authors use different words and notations. Use whatever is used in your textbook or lecture notes and everything is fine. As you have said: certainty comes with experience.

It is of course your choice to carry on in confusion and guess what a function means until you land a hit. But that will not help you. The standard approach to resolving misunderstandings is to look at definitions. You can e.g. look up what Wikipedia says about functions or accept my definition in post #20, which is basically the same, only in a less formal logical notation of what univalent means. Different types of functions have even different names and are abbreviated by different letters when their defining property or their specific context is part of the name. Notation is never unambiguous and good books have a special index for the notations they use.

I would not consider myself to be well educated, if I could only understand what a function is. I'd think it useful to grasp when a statement is bad form. My latest posting simply reflects this. For what it's worth, I did take f to be the name of a function (named: f), and took f(x) above all, to be a number and a shorthand for "Of x: square and add two", (or whatever).

If I am confused about anything, it would not be with the definition of a function, but some of the statements authors write. It is said x^2+2 is not to be referred to as a function, but is merely an expression. Here is what my book says - "Often instead of talking about function f defined by the rule f(x)=x^2+2, we shall simply say "the function X^2+2, or even "the function f(x)". Remember other notations are possible."

This shows that when I post what I think, it's based on something I've read, I've not just formulated my understanding from merely thinking about the issue.

So, the spanner in the works and the source of any misunderstanding or confusion is more to do with some of the statements authors use, which can be contested. Not confusion over what a function is. Probably true for all beginners.

Also, we should perhaps take into account what statement is seeking to do. If it's to define something or to establish equality in a mathematical sense, then that's one thing. It's another to simply give some other way to indicate which function is being talked about. In the light of this "..the function X^2+2" is probably OK. I'm thinking.

This thread is finished I think. I'd say is was more about referencing a function, rather than a thread about what is a function. Thank you for all contributions. Rich.

One more thing dawned on me. For some reason I've thought the introduction of the notation f(x) was in some way needed to get to grips with functions. But, as you know, we deal with a function when we write (say) y=2x. So, we don't need f(x) to help us understand functions, but the very fact that we use the notation f(x) indicates the equation or term involved is explicitly a function and not just any kind of equation or term.

richard9678 said:

So, we don't need f(x) to help us understand functions,

Many people rely on f(x) type notation to understand important operations on functions. The chain rule in calculus is Df(g(x)) = f'(g(x)) g'(x). Students must understand the distinction between f'(g(x)) g'(x) versus f'(x)g'(x).

Of course, others rely on the Liebnitz notation df/dy dy/dx.

Stephen Tashi said:

Many people rely on f(x) type notation to understand important operations on functions. The chain rule in calculus is Df(g(x)) = f'(g(x)) g'(x). Students must understand the distinction between f'(g(x)) g'(x) versus f'(x)g'(x).

Of course, others rely on the Liebnitz notation df/dy dy/dx.

I like the notation
$$
\left. \dfrac{d}{dx}\right|_{x_0}(f\circ g)(x)=\left. \dfrac{d}{dy}\right|_{y_0=g(x_0)}f(y)\cdot \left. \dfrac{d}{dx}\right|_{x_0}g(x)
$$
Too often are derivatives written when actually their value at a certain point is meant.