A community health researcher studies nutrient concentration in soil, modeled by \( N(x) = rac{x^3 - 8}{x - 2} \). Find \( N(3) \) in simplest form and determine whether the function is defined at \( x = 2 \).

["Community Health and Soil Science: Analyzing Nutrient Concentration via the Function ( N(x) = \frac{x^3 - 8}{x - 2} )", "In community health research, understanding environmental determinants—such as soil nutrient levels—is vital for promoting agricultural sustainability and public well-being. One key factor influencing food quality is the nutrient concentration in soil, often modeled mathematically. A recent study investigates a nutrient concentration function ( N(x) = \frac{x^3 - 8}{x - 2} ), where ( x ) represents a soil parameter (e.g., pH-adjusted nutrient availability). This article explores simplified form of the function, evaluates ( N(3) ), and examines whether ( N(x) ) is defined at ( x = 2 ).", "---", "### Simplifying the Nutrient Function", "The given model is:", "[\nN(x) = \frac{x^3 - 8}{x - 2}\n]", "Recognize that the numerator is a difference of cubes:\n[\nx^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)\n]", "Substitute this factorization into the function:", "[\nN(x) = \frac{(x - 2)(x^2 + 2x + 4)}{x - 2}\n]", "For all ( x <br/>\neq 2 ), the ( x - 2 ) terms cancel:", "[\nN(x) = x^2 + 2x + 4 \quad \ ext{for } x <br/>\neq 2\n]", "Thus, the simplified form of ( N(x) ) is ( x^2 + 2x + 4 ), but with a domain restriction: ( x <br/>\neq 2 ).", "---", "### Evaluating ( N(3) )", "Now compute ( N(3) ) using the simplified expression:", "[\nN(3) = 3^2 + 2(3) + 4 = 9 + 6 + 4 = 19\n]", "✅ Answer: ( N(3) = 19 )", "This value represents the normalized nutrient concentration at ( x = 3 ), a data point useful for assessing soil fertility and guiding community-based agricultural initiatives.", "---", "### Is ( N(x) ) Defined at ( x = 2 )?", "Although the simplified expression ( N(x) = x^2 + 2x + 4 ) appears defined at ( x = 2 ) (yielding ( 4 + 4 + 4 = 12 )), the original rational function has a removable discontinuity at ( x = 2 ).", "Since the function is undefined at ( x = 2 ) in its original form—due to division by zero—the simplified version is equivalent only for ( x <br/>\neq 2 ). Therefore, ( N(x) ) is not defined at ( x = 2 ).", "This insight reminds researchers that mathematical models must account for domain restrictions to ensure accurate interpretation—especially when informing health and farming policies.", "---", "### Conclusion", "The nutrient concentration function ( N(x) = \frac{x^3 - 8}{x - 2} ) simplifies to ( x^2 + 2x + 4 ) for ( x <br/>\neq 2 ), with ( N(3) = 19 )—a valuable measure in community nutrition studies. However, due to the undefined nature at ( x = 2 ), researchers must clearly document domain limitations. Understanding these models deepens the link between environmental data and community health outcomes.", "For public health advocates and agricultural scientists, such analyses enhance the precision of soil-based interventions, helping build resilient, nutrient-rich food systems.", "---", "Keywords: nutrient concentration, soil science, community health researcher, ( N(x) = \frac{x^3 - 8}{x - 2} ), domain restriction, simplifying rational functions, ( N(3) ), simplified function, mathematical modeling in health."]