To determine the domain and range of the transformed function $y = -\frac{1}{2}h(-3x + 9)$, let's carefully analyze how each transformation affects the original function $h(x)$.

1. Original domain and range:

• Domain: $x > -2$, i.e., $(-2, \infty)$
• Range: $-5 < y < 10$, i.e., $(-5, 10)$

---

2. Step-by-step transformations:

a) Horizontal transformation: $h(-3x + 9)$

• Replace $x$ with $-3x + 9$:
  • Solve $-3x + 9 > -2$ (because the original domain requires $x > -2$).
  • $-3x > -11$
  • $x < \frac{11}{3}$
• The transformed domain becomes $(-\infty, \frac{11}{3})$.

b) Horizontal scaling: $-3x$

• The factor $-3$ introduces a horizontal compression and a reflection about the $y$-axis. The domain interval remains $(-\infty, \frac{11}{3})$, but the reflection flips the order of values.

c) Vertical transformations:

• Multiplication by $-\frac{1}{2}$ causes:

  • Vertical compression by 1/2: The range shrinks proportionally.
  • Reflection about the x-axis: The range flips over.

• Start with the original range $-5 < y < 10$:

  • After multiplication by $-\frac{1}{2}$, the range becomes $\left(-\frac{1}{2}(10), -\frac{1}{2}(-5)\right) = (-5, 2.5)$.

---

3. Final domain and range:

• Domain: $(-\infty, \frac{11}{3})$
• Range: $(-5, 2.5)$

---

Would you like to see detailed steps of any transformation or have additional questions? Here are 5 related problems you might find interesting:

1. How do horizontal shifts affect the domain of a function?
2. What happens to the range when a function is vertically stretched or compressed?
3. How do reflections (horizontal or vertical) impact the domain and range of a function?
4. What is the effect of composite transformations (e.g., shifts combined with scaling)?
5. Can the range of a function be altered without affecting the domain? How?

Tip: Always break down transformations step-by-step, starting with the innermost operations, to avoid confusion!