Family Overview
In Plain Language
Topic 2.8 is about undoing a function: finding an inverse that reverses the input-output mapping.
- Invertible: on a specified domain, each output of $f$ comes from exactly one input (2.8.A.1).
- Reverse pairs: if $f(a)=b$, then $f^{-1}(b)=a$; tables swap $(a,b)\leftrightarrow(b,a)$ (2.8.A.2, 2.8.B.2).
- Identity check: $f(f^{-1}(x))=f^{-1}(f(x))=x$ (2.8.B.1).
- Domain/range swap: Dom$(f^{-1})=$ Range$(f)$ and Range$(f^{-1})=$ Dom$(f)$ on the invertible domain (2.8.B.2).
- Graph: reflect $y=f(x)$ over $y=x$ (2.8.B.3).
- Algebra: swap $x$ and $y$ in $y=f(x)$, then solve for $y=f^{-1}(x)$ (2.8.B.4).
- Context: domain or story limits may restrict which inverse values apply (2.8.B.5).
Why It Matters
Inverses prepare exponential $\leftrightarrow$ logarithmic work (2.9–2.11) and undo chained models from 2.7. AP items test invertibility, pair reversal, reflection, algebraic inverses, and context limits.
How Families Can Help
- Ask: "Does this graph pass the horizontal line test on the given domain?"
- Practice: reverse a table of pairs.
- Check: compose $f$ and $f^{-1}$ and get $x$.
- Sketch: reflect a simple line or ray over $y=x$.
What Not to Do
- Do not write $f^{-1}(x)=1/f(x)$ (that is reciprocal, not inverse).
- Do not claim every function has an inverse on its full domain.
- Do not forget to restrict domain when the function is not one-to-one.
- Do not ignore contextual limits (e.g., nonnegative time or price).