# One-one function

A function is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique range for each element of their domain.

Other names given to the one-one function are one to one, or injective function.

One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are surjective functions and bijective functions.

## Definition

The function is called one-one function when for every value of in the domain of the function, there will be a unique value of .

## Geometric proof

Similar to the vertical line test (VLT) for functions, there is a horizontal line test (HLT) to prove the one-one property. A function is one-one if and only if no horizontal line intersects its graph more than once.

In the graph below, the function is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.

In the graph below, the function is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.

## Properties

- The domain of f equals the range of f
^{-1}. - for every x in the domain of f and f
- The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.
- If f and g are both one-one, then f°g follows injectivity.
- If g°f is one-one, then function f is one-one, but function g may not be.
- A one-one function is either strictly decreasing or strictly increasing.
- A function that is not a one-one is considered as many-to-one.
- Parabolic functions are not one-one functions.

## Examples

Examples of one-one functions include:

- Identity function: f(x) is always injective.