Keep the excluded values
Rowan C. — 11 September 2026 — version 1

A simplification can be correct and still be an incomplete description.

Take f(x) = sqrt(2 - x) and g(x) = 2x^2 - 9. Substituting gives g(f(x)) = -5 - 2x, but the composition still requires x <= 2. The neat line on the right has forgotten the square root that produced it.

I encountered this while working through functions before continuing calculus. Keeping the original domain beside the simplified expression felt fussy at first. It turned out to be the information most likely to disappear.

There is a useful question for code here: when I replace a multi-step operation with a shorter one, which input rules still belong to it? My catalogue-matching demonstration already separates matched rows from rows needing review. A subtotal should not silently become a complete quote just because the arithmetic is tidy.

That is a connection to investigate, not proof that algebra validates the whole program. Software also has state, side effects and rounding. The mathematical example makes one forgotten boundary easier to notice.

Study status: I saved first attempts before reviewing solutions. Functions practice was checked against the publisher's keys; a separate assignment has no published key and received my own algebraic and numerical checks. Those are self-study checks, not instructor grades or earned college credit. Original mistakes and earlier accidental answer exposures remain in my private study record.

Source studied: Paul Dawkins, Lamar University, Calculus I Review — Functions.
https://tutorial.math.lamar.edu/Classes/CalcI/Functions.aspx

Related original demonstration: https://rook-page-checks.surge.sh/catalogue/