Precalculus · Unit 1: Polynomial and Rational Functions · 11 min read · Updated 2026-07-19
Change in tandem (function behavior) — AP Precalculus
AP Precalculus · Unit 1: Polynomial and Rational Functions · 11 min read
1. What a Function Is: Domain, Range, and Variables★☆☆☆☆⏱ 3 min
Most interesting quantities change **together**. Fill a vase and the water's height rises as the volume poured in rises; the two vary *in tandem*. A **function** is the precise way we pin down that pairing: it takes each input and assigns it one — and only one — output.
The input is the **independent variable** (you choose it); the output is the **dependent variable** (its value depends on the input). We write $f(x)$ to mean the output the rule $f$ assigns to the input $x$.
2. Four Representations, Image, and Preimage★★☆☆☆⏱ 3 min
The **function rule** — how each input is paired with its output — can be shown in four equivalent ways. They are four views of the *same* function, and a big part of this course is translating fluently between them.
3. Increasing and Decreasing: Comparing Outputs★★☆☆☆⏱ 3 min
To describe how two quantities change in tandem, we first ask a purely *qualitative* question: as the input increases, does the output go **up** or **down**? We answer it by directly **comparing output values** — no measure of *how fast* is needed yet (that is the next topic).
4. Building the Graph: Concavity and Zeros★★☆☆☆⏱ 4 min
A graph is just the set of input–output pairs drawn in the plane, so it shows at a glance how the two values vary together. That means a **verbal description** of how two quantities change is already enough to sketch one — you do not need a formula first.
That last distinction has a name. Describing *how fast* the output changes is the subject of Topic 1.2; here we only need the idea of the **rate of change** getting larger or smaller as the input increases.
One more feature is read straight off the picture: where the graph meets the horizontal axis.
5. AP-Style Practice★★☆☆☆⏱ 5 min
Common Pitfalls
Why: The preimage is the set of ALL inputs that map to the target output. For $f(x)=x^2$, the preimage of $9$ is $\{-3, 3\}$, not just $3$.
Why: The input $x=4$ would give two outputs, $y=2$ and $y=-2$, but a function must assign exactly one output to each input.
Why: Domain and range get swapped: the domain is the set of inputs (independent variable), the range is the set of outputs (dependent variable).
Why: $f$ is undefined at $x=2$ while $g$ is defined there, so the two functions have different domains.
Why: Increasing requires $f(a)<f(b)$ for EVERY pair $a<b$ in the interval; two matching endpoints can hide a dip in between.