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I want to define a function abstractly, rather than directly. So I want to define a function $f(r)$ as the value of $x$ for which the equation $g(x,r)=0$, for example.

I'd like to do this without words. Is there a generally accepted notation to define this function? e.g. it could look something like:

$$f(r)=Solve_x(g(x,r)=0)$$ Or something like that.

user56834
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    The equation may have more than one solution; consider as $g(x,1)=0$ the equation $(x-1)(x+1)=0$. In this case $f(1)$ is not a function. – Mauro ALLEGRANZA Apr 20 '17 at 13:54
  • I know but that would just mean that $f$ is undefined at certain values of x. That can happen with any function. – user56834 Apr 20 '17 at 14:02
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    $f(r)=y \leftrightarrow \exists ! z \ [g(z,r)=0 \land y=z]$ with an extra condition (like e.g. $f(r)=0$) or undefined when the eistence and uniqueness condition is not satisfied. – Mauro ALLEGRANZA Apr 20 '17 at 14:05
  • Can you explain why you don't want to use words? If you are doing something abstractly then communication of the idea should be key. – Ian Miller Apr 20 '17 at 14:28
  • Its often more succinct and easier to read. The same way an explicit function is easier to read than an implicit one. – user56834 Apr 20 '17 at 14:36

2 Answers2

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You could write something like

Define $f$ by

$$g(f(r),r)=0$$

but you should be careful to check that such a function actually can be defined. For example if $g(x,r)$ was $x^2+r^2-1$ then it would be unclear what $f$ was supposed to be, because there's more than one possible $x$ for each $r$.

Community
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Oscar Cunningham
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If you've told the reader at the beginning that you'll be using a certain notation (e.g. $:=$) for definitions and you've established the convention that you don't care about codomains (that a function is just its corresponding graph/relation) and there is always a unique solution to your equation, you could write something like the following:

$f:=\{(r,s)\mid g(s,r)=0\} $

I would not recommend this method. Words are used in math to make things more readable.

Mark S.
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