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The question you're working on asks to:

1. **Sketch the graph of the function** \( g(x) = 2 \sqrt[3]{x} - 3 \).
2. **Determine whether the function's inverse is also a function**.

### Step 1: Sketching the Graph of \( g(x) = 2 \sqrt[3]{x} - 3 \)

The function involves a cube root, which is different from a square root because cube roots are defined for all real numbers (both positive and negative).

- The basic shape of \( \sqrt[3]{x} \) is symmetric about the origin. It starts from the bottom left, passes through the origin, and continues to the top right.
- Multiplying by 2 vertically stretches the graph.
- Subtracting 3 shifts the graph downward by 3 units.

### Key Points to Sketch:
1. The graph will still pass through the origin, but now the point becomes \( (0, -3) \).
2. The cube root function behaves similarly for positive and negative values of \( x \). So, for negative \( x \), the graph continues smoothly into the third quadrant.
3. For large positive \( x \), the function will grow slowly but steadily.
4. For large negative \( x \), the function will approach negative infinity.

### Step 2: Determine if the Inverse is a Function

To determine if the inverse of \( g(x) \) is a function, we can apply the **horizontal line test** on the graph of \( g(x) \):

- If any horizontal line intersects the graph of \( g(x) \) at more than one point, the inverse is **not** a function.
- Since the cube root function (and its transformations) passes the horizontal line test, the inverse of \( g(x) \) will also be a function.

### Conclusion:
The inverse of \( g(x) = 2 \sqrt[3]{x} - 3 \) **is a function**.

Would you like more details on how to sketch the graph or the concept of the horizontal line test? 

Here are 5 related questions to expand on the topic:

1. What is the general form of cube root functions and their transformations?
2. How do vertical stretches affect the graph of a function?
3. What does shifting a graph downward do to its key points?
4. Can a function with a square root have an inverse that is also a function?
5. How does the horizontal line test work in determining the invertibility of a function?

**Tip:** When sketching graphs, always focus on how the transformations (stretch, shift, reflection) affect key points like intercepts.

Sketch the graph of g(x) = 2 * cube root(x) - 3 and determine if the inverse of the function is also a function.

Learn how to sketch the graph of g(x) = 2 * cube root(x) - 3 and determine if its inverse is a function using the horizontal line test. This problem involves algebraic concepts such as graphing and inverse functions. Suitable for students in Grades 10-12.

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