Part 1: Graph Transformation

The function given is: $g(x) = (x - 4)^3 + 2$

To graph this function, we start with the basic cubic function $y = x^3$ and then apply the transformations. Here's a breakdown of the transformations involved:

1. Shift 4 units to the right: The term $(x - 4)$ indicates a horizontal shift of 4 units to the right. This means every point on the graph of $y = x^3$ will move 4 units to the right.

2. Shift 2 units upwards: The constant $+2$ at the end of the equation indicates a vertical shift 2 units upwards. This means every point on the graph of $y = (x - 4)^3$ will move 2 units up.

Key Points for $y = x^3$

Let's consider a few key points on the basic cubic graph $y = x^3$ before transformations:

• When $x = -1$, $y = (-1)^3 = -1$
• When $x = 0$, $y = (0)^3 = 0$
• When $x = 1$, $y = (1)^3 = 1$

Now, we apply the transformations to these points for the function $g(x) = (x - 4)^3 + 2$:

• For $x = -1$: $g(-1) = (-1 - 4)^3 + 2 = (-5)^3 + 2 = -125 + 2 = -123$ So, the transformed point is $(-1, -123)$.

• For $x = 0$: $g(0) = (0 - 4)^3 + 2 = (-4)^3 + 2 = -64 + 2 = -62$ So, the transformed point is $(0, -62)$.

• For $x = 1$: $g(1) = (1 - 4)^3 + 2 = (-3)^3 + 2 = -27 + 2 = -25$ So, the transformed point is $(1, -25)$.

Table of Coordinates

Points on $y = x^3$	Corresponding Points on $g(x) = (x - 4)^3 + 2$
(-1, -1)	(-1, -123)
(0, 0)	(0, -62)
(1, 1)	(1, -25)

Part 2: Graphing the Function

You can graph $y = x^3$ and apply the transformations to get the graph of $g(x)$:

• Start with the basic cubic graph.
• Shift all points 4 units to the right.
• Then, shift all points 2 units upwards.

Part 3: Domain of $g(x)$

The domain of a cubic function is always all real numbers, because there are no restrictions on $x$.

Thus, the domain of $g(x)$ is: $(-\infty, \infty)$

Part 4: Range of $g(x)$

Similarly, because a cubic function can take any real value as $y$, the range of $g(x)$ is also all real numbers.

Thus, the range of $g(x)$ is: $(-\infty, \infty)$

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Would you like any further details or clarifications? Also, here are a few related questions to consider:

1. How would the graph change if the equation was $g(x) = (x + 4)^3 + 2$?
2. What would happen if the transformation shifted the graph 4 units downward instead of upward?
3. How does the graph of $g(x) = (x - 4)^3 + 2$ compare to the graph of $g(x) = - (x - 4)^3 + 2$?
4. What happens to the graph if the cubic function is compressed or stretched vertically by a factor of 2?
5. How do horizontal and vertical shifts affect the domain and range of a function?

Tip: Always remember that cubic functions have a domain and range of all real numbers unless there are restrictions like square roots or divisions by zero in the function.