# Reciprocal of a Function

## Reciprocal of a Function

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 = -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, $$^{-1}$$(x) : $$\mathbb{R}$$ → $$\mathbb{R}$$ : xx – 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.