Problem

Using the whole-interval criterion, test whether average rate of change is meaningful for \(f(x)~=~(x^2~-~1)/(x~-~1)\) on \([0,~2]\). State the original domain, inspect the entire interval, and justify the validity conclusion after simplification.

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Domain claim: x ≠ 1. Interval check: [0,2] does not contain 1. Validity: yes; the excluded input lies outside the interval.

Domain claim: x ≠ 1. Interval check: [0,2] contains 1. Validity: no; the original function has a hole inside the interval despite its computable endpoints.

Domain claim: all real x after cancellation. Interval check: [0,2] lies in the domain. Validity: yes; the simplified expression removes the hole.

Domain claim: x ≠ 1. Interval check: both endpoints 0 and 2 are allowed. Validity: yes; endpoint existence alone makes the whole-interval rate valid.

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Using the whole-interval criterion, test whether average rate of change is meaningful for \(f(x)~=~(x^2~-~1)/(x~-~1)\) on \([0,~2]\). State the original domain, inspect the entire interval, and justify the validity conclusion after simplification.

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