Family Overview

In Plain Language

Topic 2.7 is about combining functions: the output of one function becomes the input of another.

• Notation: $(f\circ g)(x)=f(g(x))$ — read "$f$ of $g$ of $x$" (2.7.A.1).
• Domain: use inputs $x$ where $g(x)$ is defined and lands in the domain of $f$.
• Evaluate: substitute, use tables/graphs, or build a formula by replacing $x$ with $g(x)$ in $f$ (2.7.A.2, 2.7.B.2).
• Not commutative: usually $f(g(x))\neq g(f(x))$ (2.7.A.3).
• Identity: composing with $I(x)=x$ returns the original function (2.7.A.4).
• Decompose: rewrite a complicated function as a composition of simpler ones (2.7.C.1).
• Transformations as composition: translations via $x+k$; dilations via $kx$ (2.7.C.2–3).

Why It Matters

Composition models chained relationships (temperature of heated water over time, then a cost rule applied to that temperature). AP items test evaluation, domain, building formulas, and decomposition.

How Families Can Help

• Inside-out evaluation: compute $g(x)$ first, then feed into $f$.
• Domain checklist: $x\in\text{Dom}(g)$ and $g(x)\in\text{Dom}(f)$.
• Compare $(f\circ g)(x)$ vs $(g\circ f)(x)$ with one numeric example.
• Decompose: "What happens to $x$ first? Then what?"

What Not to Do

• Do not assume $f(g(x))=g(f(x))$.
• Do not skip domain restrictions (especially radicals, logs, denominators).
• Do not multiply functions when composition is required ($fg$ vs $f\circ g$).
• Do not forget the identity function's role when decomposing.