1. **State the problem:** We are given the function $$g(x) = \frac{1}{2} (x + 1)^2 - 3$$ and asked to graph it, identify its domain and range. 2. **Identify the type of function:** This is a quadratic function in vertex form $$g(x) = a(x-h)^2 + k$$ where $$a = \frac{1}{2}$$, $$h = -1$$, and $$k = -3$$. 3. **Graph shape and vertex:** Since $$a = \frac{1}{2} > 0$$, the parabola opens upwards. The vertex is at $$(-1, -3)$$. 4. **Domain:** Quadratic functions are defined for all real numbers, so the domain is $$(-\infty, \infty)$$. 5. **Range:** Because the parabola opens upwards and the vertex is the minimum point, the range is all $$y$$ values greater than or equal to the vertex's $$y$$ coordinate. Thus, the range is $$[-3, \infty)$$. 6. **Summary:** - Domain: $$(-\infty, \infty)$$ - Range: $$[-3, \infty)$$ This completes the analysis and graph description of the function.