# Iterated functions

### Iterated functions

#### Introduction

• Given a function such as:
• We can call this function for x = 1:
• This is called the first iterate of x.
• We can then call the function again on the result:
• Which is the same as:
• It is called the second iterate of x.
• We can generalize the function to:
• Or even more generally:
• The functions f1(x) to fn(x) are called the iterates of x for f.
• It is also called the orbit of x
• We also use subscripts for the iterates of x:

#### Behavior of iterated functions

##### Constant
• Some functions always produce the same value:
• Produces: n 0 1 2 3 4 fn 1 1 1 1 1
##### Oscilating
• Some functions cycle between a few values:
• Produces: n 0 1 2 3 4 fn 1 -2 1 -2 1
##### Converging
• Some functions converge to a single value:
• Produces: n 0 1 2 3 4 fn 10 5.2 2.98 2.16 2
##### Random
• Some functions produce seamingly random values:
• Produces: n 0 1 2 3 4 fn 0.3 1.11 3.21 2.54 3.71
##### Expanding
• Some functions keep on growing forever:
• Produces: n 0 1 2 3 4 fn 2 4 16 256 65536

#### Graphing iterated functions

• The function converges when:
• If we plot the function, along with the line:
• Then the point where they intersect will be where the function converges:
• The function converges at xn = 2
##### Graphing the iterates
• If we start at point (x0, x0)
• We can draw a line vertically until it meets f(x)
• It will meet at the point (x0, f(x0)
• From this point we can draw a line horizontally until it meets the line: y = x
• it will meet at the point (f(x0), f(x0))
• This is the point (x1, x1)
• We can continue this process until xn = xn - 1:
##### Graphing oscilating functions
• Given the function:
• If we start at point (1, 1) and draw arrows between the two lines:
• Then we end up back where we started, at (1, 1)