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I have some algebraic function given in explicit form$-$let's take as an example $x^2-$that can attain values either $>1$ or $<1$ (let's exclude $=1$ here). I want to ascribe to a symbol $a$ the value of $x^2$ if $x^2>1$, and $1/x^2$ otherwise (so I want my $a>1$ always). So sometimes (i.e., for some values of $x$) $a=x^2$, but sometimes (for some other $x$'s) it will be $a=1/x^2$. Is there a name/symbol for such an operation?

It would seem I could do $a=\max\{x^2, 1/x^2\}$, but my real function is more complicated than $x^2$ so doing so explicitly will be messy. I could of course name the function, like $f(x)=x^2$ and then $a=\max\{f(x), 1/f(x)\}$, but this introduces a new symbol$-f(x)-$that I won't use anymore, as I'm only interested in $a$.

Question: Is there something$-$similar to $\max$, which takes the bigger of two (well, arbitrary many) values$-$that says "take the bigger from the two: value and its reciprocal"?

Note: Introducing some fancy notation, e.g. some weird brackets like $a=\uparrow x^2\uparrow$ is the same as simply introducing a function $f$.

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I don't think there is a shortcut for that. I see two options:

  • give a name to your complicated expression, e.g. call it $y$, and then write $$a:=\max(y,y^{-1})$$
  • give a name to the function: $$f(x):=\max(x,x^{-1})$$ and then call it on you expression: $a=f(\text{complicated expression})$.

In both cases you only need to write you expression once. Which one is better is up to you, it may depend on whether you will need to use that function more than once.

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