### Operations for functions: Inverse functions

### Injective functions

Injectivity

A real function #f# is called **injective** if, for all #x# and #y# in the domain of #f# with #f(x) = f(y)#, we have #x=y#.

In terms of graphs: the function #f# is injective if there is no horizontal line intersecting #f# in two (or more) points.

An important kind of injective function can be pointed out by means of the following definitions.

Monotonic functions

A real function #f# is called **increasing** if, for all #x# and #y# in the domain of #f# with #x\lt y#, we have #f(x) \lt f(y)#, and **decreasing** if, for all #x# and #y# in the domain of #f# with #x\lt y#, we have #f(x) \gt f(y)#.

A function that is either increasing or decreasing is called **monotonic**.

In the literature, what is called increasing, decreasing, and monotonic here, is also called strictly increasing, strictly decreasing, and strictly monotonic, respectively.

If the strict inequalities #\lt# and #\gt# are replaced by the weak inequalities #\le# and #\ge# in the definition, then we are talking about **weakly increasing**, **weakly decreasing**, or **weakly monotonic**.

The relevance of monotony for injectivity becomes clear from the following statement.

Injectivity for monotonic functions

If #f# is a monotonic function, then #f# is injective.

We will only prove the case where #f# is increasing. The proof for decreasing functions is similar.

Assume that #x# and #y# are points of the domain of #f# with #f(x)=f(y)#. In order to establish that #f# is injective, we need to derive #x=y# from this.

If #x\ne y#, then #x\lt y# or #x\gt y#. But #f# is increasing, so, in the first case, we have #f(x)\lt f(y)# and in the second case #f(x)\gt f(y)#, both contradicting #f(x)=f(y)#. We conclude that #x=y#, as required to prove that #f# is injective.

This can be seen in the following graph of the function #f#:

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