The reciprocal relationship of a function

*f*of X in Y is the relation noted*f*^{-1}, of Y in X, so that, for all elements of the domain of*f*, if*y*=*f*(*x*), then*x*=*f*^{-1}(*y*).- When the reciprocal relationship of a function
*f*is also a function, we call it the**reciprocal function**of*f*and we say that the function*f*is invertible. - The reciprocal of a function is not necessarily a function.
- The graph of the reciprocal relationship of a function
*f*is symmetrical to the graph of*f*in relation to the angle bisector of the first quadrant of the Cartesian plane.

### Example

Consider the function *f*: \(\mathbb{R}\) → \(\mathbb{R}\) : *x* ↦ *x* + 2; therefore, *f *\(^{-1}\)(*x*) : \(\mathbb{R}\) → \(\mathbb{R}\) : *x* ↦ *x* – 2.

### Educational Note

The term *reciprocal* is an adjective that can be applied to many mathematical objects: reciprocal relationship, reciprocal function, reciprocal proposition, etc. For this reason, it is preferable to not omit the word to which it applies in a given context.

Therefore, the reciprocal relationship of the square root function does not have the same meaning as the reciprocal function of the square root function.