Inverse

The notion of inverse in mathematical operation is simply the anti-operator.

Excerpt from Wikipedia

The following paragraph is copied from Wikipedia.

Template:AnchorInverses and composition

If f is an invertible function with domain X and codomain Y, then

$f^{-1}\left(\,f(x)\,\right)=x$, for every $x\in X$; and $f\left(\,f^{-1}(y)\,\right)=y$, for every $y\in Y.$.

Using the composition of functions, we can rewrite this statement as follows:

$f^{-1}\circ f=\operatorname {id} _{X}$ and $f\circ f^{-1}=\operatorname {id} _{Y},$

where id_X is the identity function on the set X; that is, the function that leaves its argument unchanged. In category theory, this statement is used as the definition of an inverse morphism.

Considering function composition helps to understand the notation f⁻¹. Repeatedly composing a function with itself is called iteration. If f is applied n times, starting with the value x, then this is written as fⁿ(x); so f²(x) Template:= f (f (x)), etc. Since f⁻¹(f (x)) Template:= x, composing f⁻¹ and fⁿ yields fⁿ⁻¹, "undoing" the effect of one application of f.