Questions: Classify the given function as a polynomial function, rational function, or root function, and then find the domain. Write the domain in interval notation. h(x) = (x^2 + 6) / (x^2 - 7x - 18) Classify the function h(x) = (x^2 + 6) / (x^2 - 7x - 18) Choose the correct answer below. - Polynomial function - Root function - Rational function The domain of h(x) = (x^2 + 6) / (x^2 - 7x - 18) is (Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression.)

Solution Steps

To classify the function \( h(x) = \frac{x^2 + 6}{x^2 - 7x - 18} \), we observe that it is a rational function because it is the ratio of two polynomials. To find the domain, we need to determine where the denominator is not equal to zero, as division by zero is undefined. We solve the equation \( x^2 - 7x - 18 = 0 \) to find the values of \( x \) that are excluded from the domain. The domain will be all real numbers except these values.

The given function is

\[ h(x) = \frac{x^2 + 6}{x^2 - 7x - 18} \]

Since it is expressed as a ratio of two polynomials, we classify \( h(x) \) as a rational function.

To determine the domain of \( h(x) \), we need to find the values of \( x \) that make the denominator zero. The denominator is

\[ x^2 - 7x - 18 \]

We solve the equation

\[ x^2 - 7x - 18 = 0 \]

Using the quadratic formula, we find the roots:

\[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot (-18)}}{2 \cdot 1} \]

This simplifies to

\[ x = \frac{7 \pm \sqrt{49 + 72}}{2} = \frac{7 \pm \sqrt{121}}{2} = \frac{7 \pm 11}{2} \]

Calculating the roots gives us:

\[ x = \frac{18}{2} = 9 \quad \text{and} \quad x = \frac{-4}{2} = -2 \]

Thus, the excluded values are \( x = -2 \) and \( x = 9 \).

The domain of \( h(x) \) includes all real numbers except the excluded values. In interval notation, the domain is:

\[ (-\infty, -2) \cup (-2, 9) \cup (9, \infty) \]

Final Answer