Question

What is the domain of Start Fraction f over g End Fraction, given f left-parenthesis x right-parenthesis equals x plus 8 and g left-parenthesis x right-parenthesis equals x minus 3?
(1 point)
Responses

left-parenthesis negative infinity comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma infinity right-parenthesis

left-parenthesis negative infinity comma negative 3 right-parenthesis union left-parenthesis negative 3 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma negative 3 right-parenthesis union left-parenthesis negative 3 comma infinity right-parenthesis

left-parenthesis negative infinity comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis

left-parenthesis negative infinity comma negative 8 right-parenthesis union left-parenthesis negative 8 comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma negative 8 right-parenthesis union left-parenthesis negative 8 comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis

Answer 1

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**.