Recursively Defined Functions

Recursively Defined Functions

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) = n2. 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) = 1ƒf (0) = 1ƒ1 = 1 f (2) = 2ƒf (1) = 2ƒ1 = 2 f (3) = 3ƒf (2) = 3ƒ2 = 6 f (4) = 4ƒf (3) = 4ƒ6 = 24 f (5) = 5ƒf (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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