1. **State the problem:** Find the inverse function $D^{-1}(x)$ of the function $$D(x) = -\frac{2}{15}x + 10.$$ 2. **Recall the formula for inverse functions:** To find the inverse, swap $x$ and $y$ in the equation and solve for $y$. 3. **Set $y = D(x)$:** $$y = -\frac{2}{15}x + 10.$$ 4. **Swap $x$ and $y$:** $$x = -\frac{2}{15}y + 10.$$ 5. **Solve for $y$:** Subtract 10 from both sides: $$x - 10 = -\frac{2}{15}y.$$ Divide both sides by $-\frac{2}{15}$: $$y = \frac{x - 10}{-\frac{2}{15}} = (x - 10) \times \left(-\frac{15}{2}\right).$$ 6. **Simplify:** $$y = -\frac{15}{2}x + \frac{15}{2} \times 10 = -\frac{15}{2}x + 75.$$ 7. **Conclusion:** The inverse function is $$D^{-1}(x) = -\frac{15}{2}x + 75.$$ 8. **Match with options:** This corresponds to option A. **Final answer:** $$\boxed{D^{-1}(x) = -\frac{15}{2}x + 75}.$$