Definition of a function
Two quantities are paired up. When does knowing the first one settle the second, and what do we call that?
Where you are 3 inside
What this area is about
A function pairs each input with exactly one output. That single condition separates a function from any other pairing, and it is what lets you speak of the output for an input, build a table, or draw a graph and trust it. This area sets up the idea and its vocabulary.
How these fit together
The picture. Functions as input-output rules treats a function as a machine: one input in, exactly one output out. It also shows the four ways to present a function: words, a table, a graph or an equation.
The definition. Relations and functions starts from any pairing at all, a relation, and says which relations are functions: those where no input has two outputs, whether or not a rule can be written down.
The vocabulary. Independent and dependent variables names the two roles, the input you choose and the output that follows, and writes one in terms of the other.
A path through it
The input-output machine first, then independent and dependent variables. Relations and functions can come at any point after the machine.
The rest of the branch builds on this: function notation such as f(x), domain and range, and the vertical line test for reading the one-output rule off a graph.