Domain and range
A function takes inputs and gives outputs. Which inputs does it accept, and which outputs does it ever produce?
Where you are 4 inside
What this area is about
Every function comes with two sets. The domain is the set of inputs it accepts; the range is the set of outputs it actually produces. Describing them is the first question to ask about any function, whether it arrives as a list of pairs, a graph, a formula such as \displaystyle f(x) = \sqrt{10 - 2x}, or a real situation.
How these fit together
Procedures by form. Domain and range from pairs and tables collects the inputs and outputs of a finite list. Domain and range from a graph reads how far a graph stretches left and right, down and up.
Conventions for formulas and situations. The implied domain of a formula says that when no domain is stated, it is every real number for which the formula gives a real output. A reasonable domain for a real situation shrinks that set to the inputs that make sense, such as whole numbers of people.
A path through it
Pairs and tables first, then graphs. After graphs come the implied domain and the reasonable domain in context, in either order.
With domain and range in hand you can describe one to one functions and inverses, find the domain of a rational function, and track how a transformation moves both sets.