Rewrite functions in equivalent forms to reveal and explain useful properties.
Rewrite a rational function and classify all graph features
Problem
For \(f(x)~=~(x^2~-~4)/(x~-~2)\), factor and simplify while retaining every original exclusion. Classify all holes and vertical asymptotes, and state the end-behavior asymptote.
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What this problem is really about
Record every original denominator restriction before simplifying the rational expression. Factoring reveals whether a denominator zero cancels or remains: a canceled factor creates a removable hole, while an uncanceled one can create a vertical asymptote. Keep the restriction after cancellation, use the simplified rule to locate the missing point, and read the long-run line from that rule.
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