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.