Study Guide

Learning Objectives

• Evaluate $(f\circ g)(x)$ for given values and representations (2.7.A.1–2).
• Determine domain restrictions for composites (2.7.A.1).
• Explain non-commutativity with examples (2.7.A.3).
• Use the identity function in composition (2.7.A.4).
• Construct analytic, numerical, and graphical composites (2.7.B.1–3).
• Decompose functions into compositions (2.7.C.1).
• Relate translations and dilations to composition (2.7.C.2–3).

Suggested Practice Sequence

1. Numeric evaluation: $f(2)=5$, $g(1)=2$ $\Rightarrow$ $(f\circ g)(1)=5$. 2. Symbolic: $f(x)=x^2$, $g(x)=x+3$ $\Rightarrow$ $f(g(x))=(x+3)^2$. 3. Domain: find $x$ so $g(x)$ is valid for $f$. 4. Table/graph chaining. 5. Decompose $h(x)=\sqrt{x+4}$ as $f\circ g$. 6. Show translation $f(x+k)$ as composition.

Classroom / Check-In Moves

• "Inside function first" chant.
• Side-by-side $(f\circ g)$ vs $(g\circ f)$ on one input.
• Domain two-step checklist on the board.

Misconception Watch-List

• Confusing $f\circ g$ with $f\cdot g$.
• Wrong order in composition.
• Ignoring domain of inner function output.
• Decomposition that does not recompose correctly.

Vocabulary to Own

• composite function; $(f\circ g)(x)$; inner/outer function
• domain restriction; identity function
• decompose; translation; dilation
• commutative (composition is not)

Materials Map

• Practice: evaluate and build composites
• Word: chained context models
• Mixed / multi-*: link to transformations and exponentials
• Review & practice test: timed composition fluency
• Interactive: order, domain, and formula checks