Injective functions

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.

Is the function $f(x)=-2\cdot x^4-2$ on $\mathbb{R}$ increasing, decreasing, or neither of these?

neither increasing nor decreasing

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

New example