Questions: Find the domain of the following rational function. H(x)=-5x^2/(x-2)(x+8) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain of H(x) is x ? (Type an inequality in the form x ≠. Use integers or fractions for any numbers in the expression. Use a comma to separate answers B. The domain of H(x) is the set of all real numbers

Solution Steps

To find the domain of the rational function \( H(x) = \frac{-5x^2}{(x-2)(x+8)} \), we need to determine the values of \( x \) that make the denominator zero, as these values are not included in the domain. The domain will be all real numbers except these values.

1. Set the denominator equal to zero: \((x-2)(x+8) = 0\).
2. Solve for \( x \) to find the values that are not in the domain.
3. The domain of \( H(x) \) will be all real numbers except these values.

The rational function is given by

\[ H(x) = \frac{-5x^2}{(x-2)(x+8)}. \]

To find the domain, we need to analyze the denominator, which is

\[ D(x) = (x-2)(x+8). \]

We set the denominator equal to zero to find the values of \( x \) that are not allowed in the domain:

\[ (x-2)(x+8) = 0. \]

Solving the equation gives us:

\[ x - 2 = 0 \quad \Rightarrow \quad x = 2, \] \[ x + 8 = 0 \quad \Rightarrow \quad x = -8. \]

Thus, the values that are excluded from the domain are \( x = 2 \) and \( x = -8 \).

The domain of \( H(x) \) is all real numbers except the excluded values. Therefore, we can express the domain as:

\[ \text{Domain of } H(x): \mathbb{R} \setminus \{-8, 2\}. \]

Final Answer