1. **Stating the problem:** We are given the function $$f(x) = -\frac{1}{2} x^3$$ and need to find its inverse function $$f^{-1}(x)$$. 2. **Recall the formula and rules:** To find the inverse function, we swap the roles of $$x$$ and $$y$$ and solve for $$y$$. Given $$y = -\frac{1}{2} x^3$$, the inverse satisfies $$x = -\frac{1}{2} y^3$$. 3. **Solve for $$y$$:** $$x = -\frac{1}{2} y^3$$ Multiply both sides by $$-2$$: $$-2x = y^3$$ Intermediate step with cancellation: $$\cancel{x} \times (-2) = y^3$$ 4. **Take the cube root of both sides:** $$y = \sqrt[3]{-2x}$$ 5. **Write the inverse function:** $$f^{-1}(x) = \sqrt[3]{-2x}$$ --- **Table of values for $$f(x) = -\frac{1}{2} x^3$$ from $$x = -2$$ to $$2$$:** | $$x$$ | $$f(x)$$ | |-------|----------| | -2 | $$-\frac{1}{2} (-2)^3 = -\frac{1}{2} (-8) = 4$$ | | -1 | $$-\frac{1}{2} (-1)^3 = -\frac{1}{2} (-1) = 0.5$$ | | 0 | $$-\frac{1}{2} (0)^3 = 0$$ | | 1 | $$-\frac{1}{2} (1)^3 = -0.5$$ | | 2 | $$-\frac{1}{2} (2)^3 = -4$$ | This table shows how the function values change with $$x$$.