Functions
Cost depends on how much you buy, a population on the year, a balance on time. What single idea describes every one of these dependences, and how many different shapes can it take?
Where you are 13 inside
What this area is about
A function turns each allowed input into exactly one output, and its graph is the picture of that rule. This branch builds the idea, then studies the main families of functions, each with its own formula, graph shape and kind of growth, and the ways of making new functions from old ones.
How these fit together
The idea. What a function is gives the definition, notation, domain, range and graphs. Transformations shift, flip and stretch a graph by changing the rule.
The families. Linear, quadratic, polynomial, rational, radical, piecewise and absolute value, exponential and logarithmic functions each grow or bend in a way their formula predicts.
New from old. Combining and composing functions adds, divides or chains two functions. Inverse functions run one backward, when no two inputs share an output.
Using them. Modeling with functions chooses a family and fits it to real data.
A path through it
What a function is and transformations come first, together. Linear functions come before piecewise functions and modeling. Polynomials come before rational functions, and both before exponentials. Composition and radical functions come before inverses, and inverses before logarithms; exponentials and logarithms support each other. Modeling waits for linear and exponential functions. Quadratics need only the function idea.
Next come limits, calculus and trigonometry.