Chapter 14
Special Functions
Draw a line, a parabola, or a polynomial and your pencil never leaves the paper: one rule covers every point on the curve. Now picture a parking garage that charges by the whole hour. The price holds steady for a stretch, then jumps, and no smooth curve passes through those numbers. Plenty of real quantities behave that way, turning corners, holding flat, or breaking off entirely. This chapter collects the function families built for them, and the interesting part is always what happens at the exact place where the smoothness gives out.
What You'll Explore
5 lessons.
- Radical Functions
Squaring a number is easy, but running that machine backward asks a different question: which number, multiplied by itself, gives what you started with? Here the square root becomes a function with a graph of its own, and solving an equation with the variable buried under a root raises a question about whether every answer is genuine.
- Absolute Value Equations and Graphs
Walk three blocks east, or three blocks west, and you finish the same distance from home although you went opposite ways. What does a function look like when it reports only how far and never which way? This lesson follows that question into a graph and into equations where the number of answers is itself a question.
- Floor and Ceiling
Eleven people need rides and each car holds four, and you end up calling three cars, not two and three quarters. Rounding to a whole number is something you already do by instinct. Turning that instinct into exact functions raises a question about what their graphs could possibly look like, and about which direction rounding should go.
- Rational Functions
Drive a hundred miles and your travel time depends on your speed. Halve the speed and the time doubles; keep slowing and it grows without end. One polynomial divided by another captures rules shaped like that, and their graphs do something no polynomial graph has ever done. This lesson works out what, and where.
- Piecewise Functions
Your phone plan charges one rate up to a limit and another rate past it, so no single formula covers the whole bill. Functions can be assembled from several rules, each governing its own stretch of inputs. What happens where two rules meet is the interesting part, and it may change how the earlier functions in this chapter look.