How does one express a function of a single variable and a constant?

[mesa]

How would someone go about writing a general expression for a function of a single variable and a constant? For example, if we have a function of two variables 'x' and 'y' we can use f(x,y). If we had a function of 'x' and the constant '∏' is it acceptable to write it as f(x,π)?

 

[Curious3141]

How would someone go about writing a general expression for a function of a single variable and a constant? For example, if we have a function of two variables 'x' and 'y' we can use f(x,y). If we had a function of 'x' and the constant '∏' is it acceptable to write it as f(x,π)?

Why would you want to do that? By definition, a constant does not change, so it should not influence the dependent variable. Functions like $x \rightarrow \pi x$ or $x \rightarrow \sin \pi x$ are completely adequately represented as $f(x)$ (omitting the $\pi$).

 

[mesa]

Why would you want to do that? By definition, a constant does not change, so it should not influence the dependent variable.
Functions like $x \rightarrow \pi x$ or $x \rightarrow \sin \pi x$ are completely adequately represented as $f(x)$ (omitting the $\pi$).

This is in reference to new functions where the constant, π for this particular example, is relevant. Proper notation is paramount otherwise there is no point.

On that note, and considering the confusion about the example f(x,∏), this particular representation does not seem appropriate, do you have an alternative suggestion?

 

[olivermsun]

There are lots of notations for a function $f$ in variable $x$ with some constant parameter $a$, besides just plain old $f(x, a).$ Some other examples are $f_a(x)$, $f^{(a)}(x)$, $f(x; a).$

 

[mesa]

There are lots of notations for a function $f$ in variable $x$ with some constant parameter $a$, besides just plain old $f(x, a).$ Some other examples are $f_a(x)$, $f^{(a)}(x)$, $f(x; a).$

Excellent, thank you olivermsun. Do any of those have a particular meaning or is,
$f(x, a)$ = $f_a(x)$ = $f^{(a)}(x)$ = $f(x; a).$ for all cases?

Does anyone else have anything to add?

 

[da_nang]

It depends on the context it's used in. Use a notation that doesn't cause confusion in your context and be consistent.

 

[HallsofIvy]

I have seen the notation f(x; c) used to indicate a function of the variable x which depends upon the parameter c. I think the word "parameter" is better here than "constant".

(You understand that the distinction between a "variable" and a "constant" that can take on different values is pretty slim!
f(x; c) is understood to mean a family of functions of x, each possible value of c giving a different function in that family.)

 

[mesa]

I have seen the notation f(x; c) used to indicate a function of the variable x which depends upon the parameter c. I think the word "parameter" is better here than "constant".

(You understand that the distinction between a "variable" and a "constant" that can take on different values is pretty slim!
f(x; c) is understood to mean a family of functions of x, each possible value of c giving a different function in that family.)

I have seen many examples of 'family of functions' (some even posted on PF) so this notation is good to know.

On another note, for this particular instance I am interested in notation for a function of one variable that also happens to have a single known constant (e.g. π, or e, or phi, etc.) as part of that function. If I had a function of a single variable 'x' and constant '∏' is f(x,∏) an acceptable form to describe said function?

 

[Jorriss]

On another note, for this particular instance I am interested in notation for a function of one variable that also happens to have a single known constant (e.g. π, or e, or phi, etc.) as part of that function. If I had a function of a single variable 'x' and constant '∏' is f(x,∏) an acceptable form to describe said function?

If it's just a constant like e then just don't put it.

 

[olivermsun]

If it's just a constant like e then just don't put it.

Only if there's no chance of confusion.

For example, I think writing $\log_e(x)$ and $\log_{10}(x)$ is a good idea when either or both could be used.

 

[mesa]

If it's just a constant like e then just don't put it.

I understand this sentiment, however if it were that simple I wouldn't be here. Do you think using the form f(x,some constant) is adequate?

 

[olivermsun]

I understand this sentiment, however if it were that simple I wouldn't be here. Do you think using the form f(x,some constant) is adequate?

It seems adequate. I've seen it in textbooks and published papers alike.

 

[mesa]

It seems adequate. I've seen it in textbooks and published papers alike.

Very good, thank you (again).

 

[Jorriss]

Only if there's no chance of confusion.

For example, I think writing $\log_e(x)$ and $\log_{10}(x)$ is a good idea when either or both could be used.

In this case you would write f(x) = lnx =log_e x. You're still not writing the e in the name of the function, f.

I understand this sentiment, however if it were that simple I wouldn't be here. Do you think using the form f(x,some constant) is adequate?

I prefer $f_c(x)$ to indicate that the function is only a function of x.

 

[olivermsun]

In this case you would write f(x) = lnx =log_e x. You're still not writing the e in the name of the function, f.

You are. The $e$ appears in the subscript of the function name $\log$. Anyway it's just one possible convention. The point is to distinguish between base $e$ and $10$ clearly.

 

[Jorriss]

You are. The $e$ appears in the subscript of the function name $\log$. Anyway it's just one possible convention. The point is to distinguish between base $e$ and $10$ clearly.

Hm, yeah I suppose you're right about this one. Though a comment, even here you aren't using the notation $f(x,a)$, it's still the notation $f_a(x)$.