Calculate, estimate, and interpret average rate of change for advanced function types.
Test whole-interval average-rate meaningfulness
Problem
Using the whole-interval criterion, test whether average rate of change is meaningful for \(f(x)~=~1/(x~-~2)\) on \([1,~3]\). State the domain, inspect the entire interval, and justify the validity conclusion.
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What this problem is really about
Endpoint arithmetic is not enough when a function has a domain break inside the interval. Find every excluded input and test whether the entire closed interval lies in the domain before interpreting a secant quotient. If a vertical asymptote splits the interval, the two endpoint values can exist while the requested whole-interval average remains invalid under the stated criterion.
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