#### 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 f
^{1}(x) to f^{n}(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 f ^{n}1 1 1 1 1

##### Oscilating

- Some functions cycle between a few values:
- Produces:
n 0 1 2 3 4 f ^{n}1 -2 1 -2 1

##### Converging

- Some functions converge to a single value:
- Produces:
n 0 1 2 3 4 f ^{n}10 5.2 2.98 2.16 2

##### Random

- Some functions produce seamingly random values:
- Produces:
n 0 1 2 3 4 f ^{n}0.3 1.11 3.21 2.54 3.71

##### Expanding

- Some functions keep on growing forever:
- Produces:
n 0 1 2 3 4 f ^{n}2 4 16 256 65536

#### Graphing iterated functions

- Let's start with the converging function:
- 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 x
_{n}= 2

##### Graphing the iterates

- If we start at point (x
_{0}, x_{0}) - We can draw a line vertically until it meets f(x)
- It will meet at the point (x
_{0}, f(x_{0}) - From this point we can draw a line horizontally until it meets the line: y = x
- it will meet at the point (f(x
_{0}), f(x_{0})) - This is the point (x
_{1}, x_{1}) - We can continue this process until x
_{n}= x_{n - 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)