Function notation
How do you write “the output of this function for that input” in a few symbols, and what can those symbols say?
Where you are 3 inside
What this area is about
Function notation names a function and its output in one short phrase: f(5) is the output of f when the input is 5. It is not multiplication. The notation lets you run a rule forwards or backwards, put whole expressions in for the input, and pack a sentence about a real situation, such as C(12) = 40, into a few symbols.
How these fit together
The basic procedure. Evaluating a function at a number puts a number in for the input and works out the output, and also asks the backwards question: which input gives this output?
Two extensions. Evaluating a function at an expression puts in something like x + 2 and explains why f(x + 2) is in general a different thing from f(x) + 2. Function notation in a real situation translates statements in notation into sentences with units, and back.
A path through it
Evaluate at a number first. The other two both build on it and can come in either order.
Function notation is the language for the rest of the branch: reading values from a graph, average rate of change, combining and composing functions and modeling with functions all write in it.