Functions defined by recursion (or induction) are usually discussed only after a beginning course in algebra.

They play a role both in the definition of sequences of numbers and functions usually given a more thorough treatment in calculus courses.

For example,  suppose the function $f_{ \Sigma k}: N \rightarrow N$  is defined recursively, by $f_{ \Sigma k}(0)= 0$ and $f_{ \Sigma k}(n+1)= n+ 1 +f_{ \Sigma k}(n)$ for all $n \ge 0$. Then this function  can also be characterized by $f_{ \Sigma k}(n) = \frac {n*(n+1)} 2$.

Another example was demonstrated with the factorial function, $f : N \rightarrow N$.  This function is defined by $f(0)=1$ and $f(n+1)=(n+1) \times f(n)$ for all $n \ge 0$ which can also be expressed with an elliptic notation: $f(0) = 1; f(n) = n*(n-1)* ... * 2*1$ for $n >0$.

An example of  the use of recursion to define a family of functions $f_n : R \rightarrow R$ is given in Example OW.RECF.4.

The following examples show some of the ways functions defined by recursion can be visualized with mapping diagrams.

Example OW.RECF.1 : $f: N \rightarrow N$
$f(0)=1$ and $f(n+1)=(n+1) \times f(n)$.

Example OW.RECF.2 :  $f_{ \Sigma k}: N \rightarrow N$
$f_{ \Sigma k}(0)= 0$ and $f_{ \Sigma k}(n+1)= n+ 1 +f_{ \Sigma k}(n)$ for all $n \ge 0$.

Example OW.RECF.3 :  $f: N \rightarrow N$ .
Choose $r \in R$. $f(0)= 1$ and $f(n+1)= 1 + r* f(n)$ for all $n \ge 0$.
Generalized: Choose $r \in R$. $f_a(0)= a$ and $f_a(n+1)= a + r* f(n)$ for all $n \ge 0$.

Example OW.RECF.4 :  $f_n: R \rightarrow R$ .
Choose $x \in R$. $f_0(x)= 1$ and $f_{n+1}(x)= 1 + x* f_n(x)$ for all $n \ge 0$.

Example OW.DRECF.0 : Dynamic Visualization of the Recursive Functions: Graphs and Mapping Diagrams