Is there a mathematical function that inverts a fraction if its less than 1?

Question

Simple question really: has anyone standardized a function that reverses the operators when the result is less than 1, similar to how the absolute value works for addition?
That is:

$$f(2)=2 $$ $$f(0.5)=2 $$ $$f(0.2)=5 $$ $$f(1)=1 $$

Answer 1

Are you asking if: $$f(x) = \begin{cases} x, &|x|\geq1\\ \frac{1}{x}, &|x| < 1 \end{cases}$$ Has it's own special notation, like $\log$ or $\exp$? If so, no. But, if you're just asking for the above function, there it is.

Edit: forgot some necessary conditions on $f$. The function should be:

$$f(x) = \begin{cases} x, &|x|\geq1\\ b, &|x| < 1 \end{cases}$$ Where when $|x|<1$, $x=\frac{a}{b}$ with coprime $a,b$. Note that this function isn't defined for irrational $x$ when $-1 < x <1$.

Answer 2

If its argument $x$ is assumed positive, use $\exp|\ln x|$. This can be extended to an even function on nonzero reals, viz. $\exp|\ln |x||$.

Answer 3

... similar to how abs works for additions?

Using the group isomorphism $\log: (\Bbb R^+, \cdot) \to (\Bbb R, +)$ and its inverse $\exp$ you can write that function as $$ f(x) = \exp(|\log(x)|) $$ for $x > 0$.