## Functions

A function takes an input, applies at least one rule and then produces an output.

The notation of a function is as follows:

$$\text{[Domain]}\to\text{[Co-Domain]} \text{ defined by }f(x)$$

• The domain is the collection of all input values.

• The co-domain is the collection of all possible output values.

• The range is the collection of all used output values.

• Every input should have exactly one output.

• Multiple inputs can have the same output.

• The range of a function will be a subset of the co-domain.

### Graphing

If you have the graph of your function, there is an easy way to check if something is an function.

• If you can draw a vertical line anywhere on the graph and have it intersect the graph line exactly once, then it is a function.
• If it touches the line twice, then it is not a function.
• If it never touches a line, it may still be a function, as there are functions with points removed.

An example of the graphing test.

### Composition

A functional composition of two functions $f,g$ is a function $t$ such that $t=f(g(x))$.