Problem

For \(f(x)~=~(x~+~3)/[(x~-~1)(x~+~2)]\), analyze every denominator factor and zero. State numerator matches, cancellation, the reduced formula, all discontinuities, and exclusions.

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x-1: zero 1, no numerator match, uncanceled, vertical asymptote x=1. x+2: zero -2, no numerator match, uncanceled, vertical asymptote x=-2. Reduced formula: unchanged. Exclusions: x≠-2,1.

x-1: zero 1, unmatched and uncanceled, vertical asymptote x=1. x+2: zero 2, unmatched and uncanceled, vertical asymptote x=-2. Reduced formula: unchanged. Exclusions: x≠-2,1.

x-1: zero 1, unmatched and uncanceled, vertical asymptote x=1. x+2 is omitted, so x=-2 is treated as allowed. Reduced formula: unchanged. Exclusion: x≠1.

x-1: zero 1, claimed numerator match, canceled, hole at x=1. x+2: zero -2, claimed numerator match, canceled, hole at x=-2. Reduced formula: x+3. Exclusions: x≠-2,1.

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For \(f(x)~=~(x~+~3)/[(x~-~1)(x~+~2)]\), analyze every denominator factor and zero. State numerator matches, cancellation, the reduced formula, all discontinuities, and exclusions.

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