To find the domain of the function \( \frac{f(x)}{g(x)} \), we need to determine where the denominator \( g(x) \) is not equal to zero, as division by zero is undefined.
Given:
- \( f(x) = x + 8 \)
- \( g(x) = x - 3 \)
Set the denominator \( g(x) \) equal to zero to find the restriction:
\[ g(x) = x - 3 = 0 \]
Solving for \( x \):
\[ x = 3 \]
The function \( \frac{f(x)}{g(x)} \) is defined for all real numbers except where \( g(x) = 0 \), which occurs when \( x = 3 \).
Thus, the domain of \( \frac{f(x)}{g(x)} \) is all real numbers except 3. In interval notation, this is:
\[ (-\infty, 3) \cup (3, \infty) \]
So the correct response is:
left-parenthesis negative infinity comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis.
1 answer