To determine the domain of the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] where [tex]\(f(x) = -\frac{1}{x}\)[/tex] and [tex]\(g(x) = \sqrt{3x - 9}\)[/tex], we need to analyze the domains of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] individually and then find their intersection.
1. Domain of [tex]\(f(x) = -\frac{1}{x}\)[/tex]:
- The function [tex]\(f(x) = -\frac{1}{x}\)[/tex] is defined for all [tex]\(x\)[/tex] except where the denominator is zero.
- Therefore, [tex]\(x \neq 0\)[/tex].
- This means the domain of [tex]\(f(x)\)[/tex] is all real numbers except zero:
[tex]\[
(-\infty, 0) \cup (0, \infty)
\][/tex]
2. Domain of [tex]\(g(x) = \sqrt{3x - 9}\)[/tex]:
- The square root function [tex]\(g(x) = \sqrt{3x - 9}\)[/tex] is defined when the expression inside the square root is non-negative.
- Thus, [tex]\(3x - 9 \geq 0\)[/tex].
- Solving for [tex]\(x\)[/tex], we get:
[tex]\[
3x \geq 9 \implies x \geq 3
\][/tex]
- Therefore, the domain of [tex]\(g(x)\)[/tex] is:
[tex]\[
[3, \infty)
\][/tex]
3. Domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex]:
- To find the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex], we need to find where both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are defined simultaneously.
- The domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is the intersection of the domains of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
- The domain of [tex]\(f(x)\)[/tex] is [tex]\((- \infty, 0) \cup (0, \infty)\)[/tex], and the domain of [tex]\(g(x)\)[/tex] is [tex]\([3, \infty)\)[/tex].
- The intersection of these two domains excludes any points not common to both intervals:
[tex]\[
(-\infty, 0) \cup (0, \infty) \text{ and } [3, \infty)
\][/tex]
- Since [tex]\(x = 0\)[/tex] is excluded from the domain of [tex]\(f(x)\)[/tex], we only consider the portion where [tex]\(x \geq 3\)[/tex].
4. Final Domain:
- The intersection of these intervals is [tex]\(x \geq 3\)[/tex], since for [tex]\(x\)[/tex] to be in both domains, it must satisfy [tex]\(x \geq 3\)[/tex] and [tex]\(x \neq 0\)[/tex], but here [tex]\(x = 0\)[/tex] does not concern us.
- Therefore, the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is:
[tex]\[
[3, \infty)
\][/tex]
So, the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is [tex]\( [3, \infty) \)[/tex].
