In this activity, you’ll explore what happens when a rational expression is simplified algebraically—and whether the simplification always tells the full story.[br][br][br][br]Define the original function:[br][b]f(x) := (x^2 - 1)/(x - 1)[br][br][br][/b]Use CAS to simplify:[br][b]Simplify[(x^2 - 1)/(x - 1)][/b]
What result does the CAS give you? Is this function now a simple line?[br]
Now define:[br][b]g(x) := x + 1[br][br][br][/b]Plot f(x) and g(x) on the [b]Graphics View [/b][i](down)[/i][b][br][br][br][br][br][/b]Add the point:[br][b]A = (1, 2)[/b]
What do you observe about the graphs of f(x) and g(x)? [br][br]What does this tell us about simplification?[br]
[b]Why is it important for students to graph as well as simplify?[/b][b][/b][list][*]What might students misunderstand if they only use algebra?[/*][*]How can GeoGebra help surface these issues?[/*][*]How would you structure a classroom discussion around this?[/*][/list]
You’ve seen how CAS simplifies expressions—but also how it can mask important [b]domain restrictions[/b]. This is a great opportunity to help students reason about:[list][*]Function definitions[/*][*]Discontinuities[/*][*]Limits and behavior around undefined values[/*][/list]