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Contents

Discrete Functions

Recursively Defined Functions

Problems

Problems

Recursively Defined Functions

Most of the functions we have dealt with in previous chapters have been defined explicitly: by a formula in terms of the variable. We can also define functions recursively: in terms of the same function of a smaller variable. In this way, a recursive function "builds" on itself.

A recursive definition has two parts:

  1. Definition of the smallest argument (usually f (0) or f (1) ).
  2. Definition of f (n) , given f (n - 1) , f (n - 2) , etc.

Here is an example of a recursively defined function:


We can calculate the values of this function:


f (0) = 5  
f (1) = f (0) + 2 = 5 + 2 = 7  
f (2) = f (1) + 2 = 7 + 2 = 9  
f (3) = f (2) + 2 = 9 + 2 = 11  
     

This recursively defined function is equivalent to the explicitly defined function f (n) = 2n + 5 . However, the recursive function is defined only for nonnegative integers.

Here is another example of a recursively defined function:


The values of this function are:


f (0) = 0  
f (1) = f (0) + (2)(1) - 1 = 0 + 2 - 1 = 1  
f (2) = f (1) + (2)(2) - 1 = 1 + 4 - 1 = 4  
f (3) = f (2) + (2)(3) - 1 = 4 + 6 - 1 = 9  
f (4) = f (3) + (2)(4) - 1 = 9 + 8 - 1 = 16  
     

This recursively defined function is equivalent to the explicitly defined function f (n) = n 2 . Again, the recursive function is defined only for nonnegative integers.

Here is one more example of a recursively defined function:

The values of this function are:


f (0) = 1  
f (1) = f (0) = 1ƒ1 = 1  
f (2) = f (1) = 2ƒ1 = 2  
f (3) = f (2) = 3ƒ2 = 6  
f (4) = f (3) = 4ƒ6 = 24  
f (5) = f (4) = 5ƒ24 = 120  
     

This is the recursive definition of the factorial function, F(n) = n! .

Not all recursively defined functions have an explicit definition.

The Fibonacci Numbers

One special recursively defined function, which has no simple explicit definition, yields the Fibonacci numbers:


The values of this function are:


f (1) = 1  
f (2) = 1  
f (3) = 1 + 1 = 2  
f (4) = 1 + 2 = 3  
f (5) = 2 + 3 = 5  
f (6) = 3 + 5 = 8  
f (7) = 5 + 8 = 13  
f (8) = 8 + 13 = 21  
f (9) = 13 + 21 = 34  
     

Thus, the sequence of Fibonacci numbers is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . These numbers have many interesting properties that will be studied in higher math. They recur often in mathematics and even in nature.

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