# Conventions for function notation with multiple variables

Consider the following expression:

$x\cdot a\cdot b$

In programming you could define a function $f(x,a,b)$ to assign values to all the variables. However, since I cannot remember seeing such notation being used in purely mathematical context, I am somewhat unsure whether I can do this in my math paper.

Hence, I have two questions:

1. Is it ok to define a function as $f(x,a,b)=x\cdot a\cdot b$?
2. If yes, can I refer to this function later in the paper as simply $f(x)$, omitting the other two variables? I guess what I am asking is whether $f(x,a,b)=f(x)$?
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Are $a$ and $b$ constant? If not, then you should not omit them. But if you will usually hold them constant, then one often uses them as indices; e.g., defining a family of functions $f_{ab}$ with $f_{ab}(x) = xab$. – Arturo Magidin Mar 23 '11 at 19:41
By the way, the expression $x\cdot a\cdot b$ is an expression, not an equation; you are not equating anything to anything else. – Rahul Mar 23 '11 at 19:50
@Arturo, I'm a little bit unsure about the definition but it's like you say; they are variables, assigned a constant value until changed in the next section or w/e. Not sure whether that classifies them as variables or constants. Could you expand on your answer? @Rahul thanks, fixed now! – Milosz Wielondek Mar 23 '11 at 20:04
It sounds like you want to have the freedom to assign them any value you want, but most of the time they will be fixed in practice. If this is the case, then you have two options: if the values of $a$ and $b$ will be clear from context, and you will never mix two of these functions with different values of $a$ and $b$ (e.g., you will never consider the function $x\mapsto xab$ and $x\mapsto xcd$ at the same time), then just use $f(x)=xab$. If it is important to keep track of $a$ and $b$, or you will use several at the same time, then I suggest using the subindices, $f_{ab}=xab$. – Arturo Magidin Mar 23 '11 at 20:08
@Arturo: You are correct, that is what I want. More precisely, I would like to be able to explicitly state $a$ and $b$ when they are assigned specific values, e.g. $f(x,2,\sqrt{3})=2\sqrt{3}x$, but at other times to be able to refer to that function as only $f(x)$ then assuming that $a$ and $b$ don't hold any value. Ideas? – Milosz Wielondek Mar 23 '11 at 20:17

For given values $a$ and $b$, we define the function $f_{ab}$ by $f_{ab}(x)=abx$; for example, $f_{2,\sqrt{3}}(x) = 2\sqrt{3}x$. When $a$ and $b$ are fixed and understood from context, we will often omit them and simply write $f(x)$ instead of $f_{ab}(x)$.