Lesson Plan

• Describe why a function must be one‑to‑one to possess an inverse.
• Apply the horizontal line test to decide if a function fails to be one‑to‑one.
• Explain how restricting the domain can restore invertibility.
• Analyse given functions and justify whether they have an inverse.

• Projector and screen
• Whiteboard and markers
• Graphing calculator or graphing software
• Handout with function examples and diagrams
• Worksheet for horizontal‑line‑test practice
• Sticky notes for exit tickets

Begin with a quick question: “If you could ‘undo’ any operation, what would that look like?” Connect this to the idea of inverse functions covered last lesson. State that today students will learn how to recognise when a function cannot be undone and what to do about it.

1. Do‑now (5'): Write one example of a function and its inverse on a sticky note.
2. Mini‑lecture (10'): Review bijective definition, introduce the horizontal line test with a parabola diagram.
3. Guided practice (12'): In pairs, use the handout to apply the horizontal line test to $x^{2}$, $\sin x$, and $\lfloor x\rfloor$; record reasons for failure.
4. Class discussion (8'): Share findings; emphasise “two $x$ values → one $y$” as the key issue.
5. Domain‑restriction activity (10'): Restrict $x^{2}$ to $x\ge0$, sketch the new graph, and derive $f^{-1}(y)=\sqrt{y}$.
6. Exit ticket (5'): Write one concise reason a function may not have an inverse.

Recap that a function must be one‑to‑one, verified by the horizontal line test, and that domain restriction can create an inverse. Collect exit tickets to gauge understanding, and assign the worksheet for homework, asking students to find a real‑world situation where restricting a domain is necessary.