Question

find the domain of the following rational function.\\( h(x) = \frac{-9x^2}{(x - 2)(x + 5)} \\)\\( \quad \\)\\( \quad \\)select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\( \quad \\)\\( \bigcirc \\) a. the domain of \\( h(x) \\) is \\( \\{ x \mid \square \\} \\).\\( \quad \\)\\( \quad \\)\\( \quad \\) (type an inequality in the form \\( x \
eq \\). use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\\( \quad \\)\\( \bigcirc \\) b. the domain of \\( h(x) \\) is the set of all real numbers.

Explanation:

Step1: Identify Denominator

The function is \( H(x) = \frac{-9x^2}{(x - 2)(x + 5)} \). The denominator is \( (x - 2)(x + 5) \).

Step2: Find Zeroes of Denominator

Set denominator equal to zero: \( (x - 2)(x + 5) = 0 \).
Using zero - product property, \( x - 2 = 0 \) or \( x + 5 = 0 \).
Solving \( x - 2 = 0 \) gives \( x = 2 \). Solving \( x + 5 = 0 \) gives \( x=-5 \).

Step3: Determine Domain

For a rational function, the domain excludes values that make the denominator zero. So \( x
eq2 \) and \( x
eq - 5 \).

Answer:

A. The domain of \( H(x) \) is \( \{x\mid x
eq2, x
eq - 5\} \)