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