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To determine the domain of \(\left(\frac{g}{f}\right)(x)\), we need to analyze the domains of \(f(x)\) and \(g(x)\), as well as any restrictions imposed by the division of functions.

1. **Identify \(f(x)\):**
   \[
   f(x) = \frac{5x}{x - 1}
   \]
   For \(f(x)\), the denominator \(x - 1\) must not be zero, so \(x \neq 1\).

2. **Identify \(g(x)\):**
   \[
   g(x) = \sqrt{2x + 6}
   \]
   For \(g(x)\), the expression inside the square root must be non-negative:
   \[
   2x + 6 \geq 0 \implies x \geq -3.
   \]

3. **Domain of \(\frac{g(x)}{f(x)}\):**
   Since \(\left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)}\), we combine the restrictions from \(f(x)\) and \(g(x)\):
   - \(x \geq -3\) (from \(g(x)\))
   - \(x \neq 1\) (from \(f(x)\))

   Thus, the domain of \(\left(\frac{g}{f}\right)(x)\) is:
   \[
   [-3, 1) \cup (1, \infty)
   \]

4. **Match with Answer Choices:**
   The correct answer is:
   - **Option (c):** \([-3, 1) \cup (1, \infty)\)

Would you like a detailed explanation of any step?

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Here are five related questions to expand your understanding:

1. How do you determine the domain of functions with square roots?
2. What are common restrictions to consider in rational functions?
3. How do you determine the domain of combined functions like \(\frac{g}{f}\)?
4. What is the effect of square roots and denominators on the domain?
5. Why is it important to consider each part of a composite function separately?

**Tip:** Always check for both square roots and denominators when finding the domain of a fraction.

Misalkan f(x) = \(\frac{5x}{x-1}\) dan g(x) = \sqrt{2x + 6}. Tentukanlah domain dari fungsi \(\frac{g}{f}\)(x).

Determine the domain of the composite function \(\frac{g}{f}\)(x) where f(x) = \(\frac{5x}{x-1}\) and g(x) = \sqrt{2x + 6}. Explore the restrictions imposed by rational and square root functions, suitable for Grades 11-12.

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