To graph the function $g(x) = (x + 3)^3 - 1$, we can apply a sequence of transformations to the basic cubic function $f(x) = x^3$. Let's break down the steps:

Step-by-step Transformations:

1. Start with the basic function:
   The parent function is $f(x) = x^3$, which has an "S"-shaped curve passing through the origin (0, 0).

2. Horizontal shift (left by 3 units):
   The term $(x + 3)$ shifts the graph of $f(x) = x^3$ to the left by 3 units. So, every point on the graph of $f(x) = x^3$ moves left by 3 units.
   Now, the graph passes through the point (-3, 0) instead of (0, 0).

3. Vertical shift (down by 1 unit):
   The $-1$ at the end shifts the graph down by 1 unit. So, the point that was at (-3, 0) moves to (-3, -1), and the entire graph is shifted down by 1 unit.

Domain and Range:

• Domain: The domain of $g(x) = (x + 3)^3 - 1$ is the set of all possible $x$-values for which the function is defined. Since there are no restrictions on the value of $x$ in a cubic function, the domain is: $\text{Domain} = (-\infty, \infty)$

• Range: The range is the set of all possible $y$-values that the function can take. Since cubic functions can take any real value and this function is simply a shifted version of the basic cubic, the range is also: $\text{Range} = (-\infty, \infty)$

Summary of Transformations:

• Left shift by 3 units.
• Down shift by 1 unit.

Graph Description:

The resulting graph will be an "S"-shaped curve passing through the point (-3, -1), with the general cubic shape preserved. The function grows indefinitely in both directions, so it will continue to rise as $x \to \infty$ and fall as $x \to -\infty$.

---

Would you like me to provide a detailed graph or further clarification?

Here are 5 related questions:

1. How do horizontal and vertical stretches affect the graph of a cubic function?
2. What is the effect of reflecting a cubic function over the x-axis or y-axis?
3. Can you describe how combining multiple transformations (e.g., reflections and shifts) affects the function's graph?
4. What happens to the graph of $g(x) = x^3$ when compressed vertically or horizontally?
5. How would you determine the inverse of a cubic function?

Tip: When transforming graphs, always apply shifts before stretches or reflections to simplify the process.