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If I have a function family paramterised with a parameter $a$ such that $f(x)=ax$ can I have it such that we can determine the value of each function $f$ at each $a$?

This would give me $f(a)=a^2$ which is misleading. Is this just a symptom of having a 'varying' function denoted by a different symbol for each $a$?

Would it better to use the arrow notation with some dummy variable to specify that we take $x$ to $ax$ and such that applying the same function to $a$ would yield the value $a^2$?

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    Write the functions in the form $f_a(x)=ax.$ Then $f_a(a)=a^2$ for each $a$. – Fred Nov 03 '22 at 08:39
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    You say you want to describe a family of functions with a varying parameter $a$; in that case you should notate each function belonging to that family with two arguments: like $f(a,x)$ or $f_a(x)$. Writing it as just $f(x)=ax$ notates a function $f$ with only one independent variable $x$ and $a$ would be considered a fixed constant. – Prasun Biswas Nov 03 '22 at 08:41
  • Oh... I see. If it's a family of functions then, as Fred points out, you must distinguish between one $f(x) =ax$ another $f(x) =bx$. And the way we conventionally do that is as a subscript. $f_a(x) =ax$ and $f_b(x) = bx$. Then it's not ambiguous to point out $f_a(a) = a^2$ and $f_k(k)=k^2$ and "for each $a$" that $f_a(a) = a^2$. The thing is, its clear that as we are inputting the exact indext of the function this is an isolated evaluate an one point. (Note: If we are told $f_a(a)=a^2$ we don't know why that is true. Just that we are told it is. – fleablood Nov 03 '22 at 09:28
  • @MorganRodgers I apologise but from my understanding, if $a$ is not previously specified as having a specific value, then it is a paramater? In this case the symbol $f$ is 'variable'. Is it more just that it is an abuse of notation to have $f$ as a variable symbol? –  Nov 03 '22 at 09:36
  • @PrasunBiswas In what way do you mean fixed constant, it has a 'specific' value because if it does not then a 'fixed constant' without a specific value is just a parameter again. –  Nov 03 '22 at 09:38
  • @Fred is this simply for clarity or does writing $f(x)=ax$ imply $a$ has to have a specific value? –  Nov 03 '22 at 09:47
  • @MorganRodgers so $f$ denotes only one function, for which there is a particular, yet unnamed number that defines it? That makes sense. –  Nov 03 '22 at 11:58
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    Does this answer your question? What is the difference between writing $f$ and $f(x)$? – Mauro ALLEGRANZA Nov 08 '22 at 10:07
  • $f(a)$ is the value of function $f$ for input $a$. – Mauro ALLEGRANZA Nov 08 '22 at 10:08

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  1. What is a "family" ? Let's first take a step back and recall what it is. Let $E$ and $I$ two sets. A family of elements of $E$ indexed by $I$ is just an application :$$\xi:I\to E$$ $$i\mapsto \xi(i)$$ For example , a sequence $u$ of real numbers is just an application from $\Bbb N$ to $\Bbb R$. That we designate by $u_n$ the real number $u(n)$ by habit, does not change anything. We also write $u=(u_n)_{n\in \Bbb N}$.

  2. A family $\xi$ is also written $(\xi_i)_{i\in I}$

  3. Here, our set $E$ is the set of functions from $\Bbb R$ to $\Bbb R$, $\mathcal {F}(\Bbb R, \Bbb R$). And we have a family indexed by $I=\Bbb R$, $(f_a)_{a\in \Bbb R}$ defined by $$f_a:\Bbb R\to \Bbb R$$ $$x\mapsto ax$$ for any $a\in \Bbb R$

  4. Once this is understood, we can answer the question : let $a\in \Bbb R; $ what is $f_a(a)$?

  5. What you write after (for example, "This would give me $f(a)=a^2$") is interesting: so be specific.

Stéphane Jaouen
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  • Is it entirely wrong to have $f$ as a 'variable' symbol with a parameter $a$ or is it simply for notations sake that we do this, obviously in science backgrounds we use parameters like 'mass' and write $F(t)=mg(t)$ where $m$ is a parameter? I understand it causes issues but usually from context its clear if $f$ denotes a specific function and then definitions like $f(a)=a^2$ have their usual meaning. –  Nov 03 '22 at 09:41
  • I know there are posts on MSE but I don't know where exactly on $:=$ and $=$; at the limit, you could write $f(x):=ax$ and $f(a)=a^2$, under cover of the approval of the other MSE members :). But it's still difficult to talk about family without indexing. – Stéphane Jaouen Nov 03 '22 at 09:46
  • I guess it can be done, but it's unclear so better to index. Thanks, but physisics dont care and will express the function very generally without it. –  Nov 03 '22 at 09:48
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    At their own risk, ["A leurs risques et périls", as we say in France :) ] – Stéphane Jaouen Nov 03 '22 at 09:50