Factor numerator: \(x^2 - 4 = (x - 2)(x + 2)\), so \(f(x) = rac{(x - 2)(x + 2)}{x - 2} = x + 2\) for \(x

["Understanding the Factor Numerator in Rational Functions: Simplifying (f(x) = \frac{x^2 - 4}{x - 2})", "Factor numerators play a crucial role in simplifying rational functions and uncovering important properties such as holes, asymptotes, and domain restrictions. One classic example is:", "[\nf(x) = \frac{x^2 - 4}{x - 2}\n]", "At first glance, this function appears straightforward—but careful analysis reveals how factoring reveals underlying behavior.", "### Factor the Numerator", "Start by factoring the numerator, a difference of squares:", "[\nx^2 - 4 = (x - 2)(x + 2)\n]", "So the function becomes:", "[\nf(x) = \frac{(x - 2)(x + 2)}{x - 2}\n]", "### Simplify the Rational Expression", "Since (x - 2) appears in both the numerator and denominator (and assuming (x <br/>\ne 2), where the denominator is zero), we can simplify the expression to:", "[\nf(x) = x + 2, \quad \ ext{for } x <br/>\ne 2\n]", "### The Factor Numerator Reveals the Simplification", "The factor ( (x - 2) ) is the factor numerator—the polynomial shaved off during simplification. While the expression simplifies to (x + 2), the original function is undefined at (x = 2), where the denominator becomes zero. This indicates a removable discontinuity or hole at (x = 2), not a vertical asymptote.", "### Key Takeaways:", "- Factor numerators enable simplification of rational expressions.\n- The factor (x - 2) cancels, but must not be zero.\n- The simplified form (x + 2) approximates (f(x)) near all (x <br/>\ne 2), but (f(x)) is undefined at (x = 2).\n- Understanding factor numerators helps identify holes rather than just asymptotes.", "In summary, recognizing and factoring the numerator is essential in algebra for proper function analysis and accurate graphing. The factor (x - 2) is key to revealing the hole, making (f(x) = x + 2) the simplified version excluding the undefined point."]