Study Guide
Learning Objectives
- Decide whether $f$ is invertible on a given domain (2.8.A.1).
- Find inverse input-output pairs from mappings and tables (2.8.A.2, 2.8.B.2).
- Verify $f\circ f^{-1}=f^{-1}\circ f=$ identity (2.8.B.1).
- Swap domain and range for $f^{-1}$ (2.8.B.2).
- Reflect graphs over $y=x$ to obtain $f^{-1}$ (2.8.B.3).
- Find algebraic inverses by swapping and solving (2.8.B.4).
- Apply contextual domain restrictions to inverses (2.8.B.5).
Suggested Practice Sequence
1. Horizontal line test / unique inputs for outputs. 2. Reverse table pairs; write $f^{-1}$ values. 3. Compose $f$ and $f^{-1}$ to check identity. 4. Reflect a graph over $y=x$. 5. Algebra: $y=2x+3 \rightarrow x=(y-3)/2 \rightarrow f^{-1}(x)=(x-3)/2$. 6. Restrict $f(x)=x^2$ to $x\ge 0$ so $f^{-1}(x)=\sqrt{x}$.
Classroom / Check-In Moves
- Pair cards: $(2,5)$ becomes $(5,2)$ for the inverse.
- "Reciprocal vs inverse" error hunt.
- Domain-restriction gallery for $x^2$, absolute value, etc.
Misconception Watch-List
- $f^{-1}=1/f$.
- Claiming $x^2$ is invertible on $\mathbb{R}$.
- Forgetting to swap domain and range.
- Solving incorrectly after swapping $x$ and $y$.
Vocabulary to Own
- inverse function; invertible; one-to-one
- $f^{-1}$; identity function
- horizontal line test; reflection over $y=x$
- domain restriction; contextual restriction
Materials Map
- Practice: invertibility, pairs, algebra, reflection
- Word: undo temperature, pricing, or conversion models
- Mixed / multi-*: link to composition (2.7) and exponentials
- Review & practice test: timed inverse fluency
- Interactive: definition and process checks