Let&nbsp;\(\mathbb{Z}\) denote the set of all integers and define \(\mathrm{f} : \mathbb{Z} \rightarrow \mathbb{Z}\) as follows:
\[
\mathrm{f}(x) = 
\begin{cases} \frac{x}{2} & \text{if}\hspace{0.5em} x \text{ is even;} \\ 
 3x + 1 & \text{if}\hspace{0.5em} x \text{ is odd.} 
\end{cases}
\]
Notice that the rule that defines \(\mathrm{f}\) is not a single algebraic expression, but nevertheless, it gives a unique value for the output \(\mathrm{f}(x)\) for every input \(x\) taken from the domain \(D_f = \mathbb{Z}\).
This example is also very interesting in that it is not so direct to obtain the range of \(\mathrm{f}\) by sketching the graph of \(y = \mathrm{f}(x)\). Indeed, you don't even get a continuous curve. Instead, you get a set of dots \((x, y)\), where&nbsp;\(x\) and \(y\) are integers.
One wonders whether \(R_f = \mathbb{Z}\), i.e., whether every integer \(y \in \mathbb{Z}\) is the image of some integer \(x\) under \(\mathrm{f}\)? But this question is not difficult to answer.
Take any random integer \(y\). Then&nbsp;\(y = \frac{2y}{2} = \mathrm{f}(2y)\) based on the fact that&nbsp;\(2y\) is certainly an even integer and the very definition of \(\mathrm{f}\).
Hence, we can be certain that&nbsp;\(R_f = \mathbb{Z}\).
The above function \(f\) is known as the Collatz function. Related to this function is a famous open problem in number theory -- The Collatz Conjecture -- which Paul Erdos once commented that "Mathematics is not ready for it." The Collatz Conjecture states that for every input \(n \in \mathbb{Z}^+\), there exists an \(m \in \mathbb{Z}^+\) such that \[
f^{(m)}(n) = 1.
\] Here, \(f^{(m)}\) stands for the \(m\)-fold composition of \(f\), a concept we will see in our ensuing development.

Let \(\mathbb{Z}\) denote the set of all integers and define \(\mathrm{f} : \mathbb{Z} \rightarrow \mathbb{Z}\) as follows:

\[
\mathrm{f}(x) = 
\begin{cases} \frac{x}{2} & \text{if } x \text{ is even;} \\ 
                              3x + 1          & \text{if } x \text{ is odd.} 
\end{cases}
\]

Notice that the rule that defines \(\mathrm{f}\) is not a single algebraic expression, but nevertheless, it gives a unique value for the output \(\mathrm{f}(x)\) for every input \(x\) taken from the domain \(D_f = \mathbb{Z}\).

This example is also very interesting in that it is not so direct to obtain the range of \(\mathrm{f}\) by sketching the graph of \(y = \mathrm{f}(x)\). Indeed, you don't even get a continuous curve. Instead, you get a set of dots \((x, y)\), where \(x\) and \(y\) are integers.

One wonders whether \(R_f = \mathbb{Z}\), i.e., whether every integer \(y \in \mathbb{Z}\) is the image of some integer \(x\) under \(\mathrm{f}\)? But this question is not difficult to answer.

Take any random integer \(y\). Then \(y = \frac{2y}{2} = \mathrm{f}(2y)\) based on the fact that \(2y\) is certainly an even integer and the very definition of \(\mathrm{f}\).

Hence, we can be certain that \(R_f = \mathbb{Z}\).

The above function \(f\) is known as the **Collatz function**.  Related to this function is a famous open problem in number theory -- The Collatz Conjecture -- which Paul Erdos once commented that "Mathematics is not ready for it." 

The Collatz Conjecture states that for every input \(n \in \mathbb{Z}^+\), there exists an \(m \in \mathbb{Z}^+\)  such that 
\[
f^{(m)}(n) = 1.
\]
Here, \(f^{(m)}\) stands for the \(m\)-fold composition of \(f\), a concept we will see in our ensuing development.

Functions

International Baccalaureate Higher Level Mathematics

An illustration of how a function is defined using a rule

An illustration of how a function is defined using a rule