What a function is
Price depends on weight, distance depends on time, height depends on age. What exactly does it mean for one quantity to depend on another, and how do we write and picture that?
Where you are 5 inside
What this area is about
A function is a rule that gives each allowed input exactly one output. That one condition is what lets you speak of the output for an input, write it as f(x), and draw a graph you can trust. This area sets up the idea, the notation, the sets of inputs and outputs, and the picture.
How these fit together
The definition. Definition of a function is an area of its own: functions as input and output rules, relations and which of them are functions, and independent and dependent variables.
The notation. Function notation writes the output of f at the input x as f(x), evaluates it at numbers and at expressions, and reads it in real situations.
Inputs and outputs. Domain and range are the inputs a function accepts and the outputs it produces, found from pairs, graphs, formulas and situations.
The picture. Graphs of functions plot every pair (x, f(x)) and read values, stories and landmarks off the result. Testing graphs and equations for functions checks, without listing every point, whether some input has two outputs.
A path through it
The definition first, then notation. After that, domain and range and graphs of functions can come in either order, and the vertical line test follows graphs.
Next comes every family of functions: linear, quadratic and exponential functions, and new functions built from old ones, such as inverses.