Chapter 2
Functions and Their Graphs
Punch a code into a vending machine and a snack drops. Enter the gigabytes you used and a phone plan returns a bill. You have written rules like these for years, and you have always used them one way: put a number in, read a number out. That habit hides something. A rule is not only a set of steps; it is an object with a shape of its own, and with behavior you can describe before you compute anything. Once a function is a thing and not just a calculation, what questions can you ask about it?
What You'll Explore
6 lessons.
- Relations and Functions
A circle drawn on a page is one curve, but whether it counts as a function turns out to depend on something that is not in the picture at all. This lesson goes underneath the definition you already know, and asks how much of a function is really contained in its formula.
- Function Notation and Evaluation
Writing f of x feels like nothing more than a label for an answer. What is actually allowed to sit inside those parentheses, and does it have to be a number at all? This lesson takes the notation seriously, and tests a tempting assumption about what happens when two inputs are added.
- Graphs of Functions
Plot five points and join them with a smooth curve. How do you know the real graph does what your pencil did between those points? This lesson takes that question seriously, and looks at what a rule can be made to tell you about its own picture before a single point is plotted.
- Transformations of Graphs
You have met a stack of transformation rules, each with a direction to memorize, and some of them stubbornly run the opposite way to what you expect. What if every one of them is the same rule in disguise? This lesson hunts for that one rule, and for what happens when two moves are combined.
- Composition of Functions
Numbers can be added, and addition obeys laws you can state and prove. Chaining one function into another is also an operation, on objects that are not numbers at all. Which of the familiar laws survive that jump, and which quietly fail? That is the question this lesson settles.
- Inverse Functions
Every operation you have run forward raises the same question: can it be undone? Some functions reverse cleanly and others refuse, and telling the two apart is not obvious from a formula. Here you will look at what undoing really demands of a function, and at the mark it leaves on the graph.