Question

Find the domain of the following rational function.
g left parenthesis x right parenthesis equals StartFraction negative 7 x squared Over left parenthesis x minus 3 right parenthesis left parenthesis x plus 8 right parenthesis EndFraction
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The domain of​ g(x) is restricted to
  
enter your response here.
​(Type your answer in interval​ notation.)
B.
The domain of​ g(x) has no restrictions.

Answer 1

To determine the domain of the rational function \( g(x) = \frac{-7x^2}{(x - 3)(x + 8)} \), we need to identify any values of \( x \) that would make the denominator equal to zero, as these values cannot be included in the domain.

1. The denominator is \( (x - 3)(x + 8) \).
2. Set the denominator equal to zero to find the values we need to exclude:
\[
(x - 3)(x + 8) = 0
\]
This gives us two equations:
- \( x - 3 = 0 \) → \( x = 3 \)
- \( x + 8 = 0 \) → \( x = -8 \)

3. Thus, the values \( x = 3 \) and \( x = -8 \) are not in the domain of \( g(x) \).

4. Since these are the only restrictions, the domain of \( g(x) \) in interval notation is:
\[
(-\infty, -8) \cup (-8, 3) \cup (3, \infty)
\]

So, the correct answer is:

A. The domain of \( g(x) \) is restricted to
\[
(-\infty, -8) \cup (-8, 3) \cup (3, \infty)
\]