Classify the given function as a polynomial function, rational function, or root function, and then fi \( f(x)=\sqrt{x^{2}-3 x-18} \)

To classify the function \( f(x) = \sqrt{x^{2} - 3x - 18} \), we need to analyze its structure. 1. **Polynomial Function**: A polynomial function is a function that can be expressed in the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are constants and \( n \) is a non-negative integer. The expression inside the square root, \( x^2 - 3x - 18 \), is a polynomial, but the entire function \( f(x) \) is not a polynomial because of the square root. 2. **Rational Function**: A rational function is a function that can be expressed as the ratio of two polynomials, i.e., \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials. The function \( f(x) \) does not fit this definition since it involves a square root rather than a ratio of polynomials. 3. **Root Function**: A root function is a function that involves a root (such as a square root, cube root, etc.) of a polynomial. Since \( f(x) \) is defined as the square root of the polynomial \( x^2 - 3x - 18 \), it fits the definition of a root function. Based on this analysis, we classify the function \( f(x) = \sqrt{x^{2} - 3x - 18} \) as a **root function**.