1. **State the problem:** Determine the domain and range of the parabola given by $$f(x) = 3x^2 - 24x + 47$$. 2. **Recall the domain of a quadratic function:** The domain of any quadratic function is all real numbers, since you can plug any real number into $$x$$. 3. **Find the vertex to determine the range:** The vertex form of a parabola $$f(x) = ax^2 + bx + c$$ has vertex at $$x = -\frac{b}{2a}$$. Calculate: $$x = -\frac{-24}{2 \times 3} = \frac{24}{6} = 4$$ 4. **Find the corresponding $$y$$ value at the vertex:** $$f(4) = 3(4)^2 - 24(4) + 47 = 3 \times 16 - 96 + 47 = 48 - 96 + 47 = -1$$ 5. **Determine the parabola's direction:** Since $$a = 3 > 0$$, the parabola opens upward, so the vertex is a minimum point. 6. **Write the domain and range:** - Domain: all real numbers - Range: $$f(x) \geq -1$$ because the minimum value is $$-1$$ at $$x=4$$. **Final answer:** Domain is all real numbers. Range is $$f(x) \geq -1$$.