given the pair of functions f and g, sketch the graph of y = g(x) by starting
with the graph of y = f(x) and using transformations. Track at least three points of your choice
through the transformations. State the domain and range of g.
21. f(x) = x^3, g(x) = (x + 2)^3 + 1

To graph \( y = g(x) \) using transformations of \( y = f(x) \), we will start with the base function \( f(x) = x^3 \) and apply the necessary transformations to obtain \( g(x) = (x + 2)^3 + 1 \).

### Steps of Transformation:

1. **Start with \( f(x) = x^3 \):**
   - This is the basic cubic function. Key points on the graph are:
     - \( (-1, -1) \)
     - \( (0, 0) \)
     - \( (1, 1) \)

2. **Horizontal Shift Left by 2 Units:**
   - The function \( g(x) = (x + 2)^3 \) is obtained by shifting \( f(x) \) horizontally to the left by 2 units.
   - New points after transformation:
     - \( (-1, -1) \) → \( (-3, -1) \)
     - \( (0, 0) \) → \( (-2, 0) \)
     - \( (1, 1) \) → \( (-1, 1) \)

3. **Vertical Shift Up by 1 Unit:**
   - The final function \( g(x) = (x + 2)^3 + 1 \) is obtained by shifting \( (x + 2)^3 \) vertically upward by 1 unit.
   - New points after transformation:
     - \( (-3, -1) \) → \( (-3, 0) \)
     - \( (-2, 0) \) → \( (-2, 1) \)
     - \( (-1, 1) \) → \( (-1, 2) \)

### Domain and Range of \( g(x) \):

- **Domain:** The domain of \( g(x) \) is all real numbers since the cubic function \( x^3 \) is defined for all \( x \). Hence, the domain is \( (-\infty, \infty) \).
  
- **Range:** The range of \( g(x) \) is also all real numbers because as \( x \to \infty \) or \( x \to -\infty \), \( g(x) \) also tends to \( \infty \) or \( -\infty \) respectively. Therefore, the range is \( (-\infty, \infty) \).

### Sketch:

1. Start with the graph of \( y = f(x) = x^3 \).
2. Shift the entire graph left by 2 units.
3. Shift the resulting graph up by 1 unit.

### Key Points Tracked:

1. Original point \( (-1, -1) \) becomes \( (-3, 0) \).
2. Original point \( (0, 0) \) becomes \( (-2, 1) \).
3. Original point \( (1, 1) \) becomes \( (-1, 2) \).

Do you want to see the graphical representation or have any questions? 

Here are five related questions:

1. How do horizontal and vertical transformations affect the graph of a function?
2. Can you describe the transformation process for a quadratic function instead of a cubic one?
3. What happens to the domain and range if the cubic function has a vertical stretch or compression?
4. How do transformations change if there is a reflection involved?
5. How would you determine the inverse of the function \( g(x) \)?

**Tip:** Always start with the innermost transformation (closest to \( x \)) when applying multiple transformations to a function.

Given the pair of functions f(x) = x^3 and g(x) = (x + 2)^3 + 1, sketch the graph of y = g(x) by starting with the graph of y = f(x) and using transformations. Track at least three points of your choice through the transformations. State the domain and range of g.

Learn how to graph cubic functions by using transformations. This example focuses on transforming f(x) = x^3 into g(x) = (x + 2)^3 + 1 by applying horizontal and vertical shifts. Follow step-by-step transformations and track key points. Suitable for students in Grades 10-12.

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