Starting with the graph of a basic​ function, graph the following function using the techniques of​ shifting, compressing,​ stretching, and/or reflecting. Find the domain and range of the function.

g left parenthesis x right parenthesis equals left parenthesis x plus 3 right parenthesis cubed minus 1

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 \( x \to -\infty \).

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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.

Starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Find the domain and range of the function.
g(x) = (x + 3)^3 - 1

Learn how to graph the cubic function g(x) = (x + 3)^3 - 1 using techniques like shifting and reflecting. This detailed guide explains transformations, domain, and range for high school students in Grades 9-12.

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