The graph shown is a translation of the function $y = \sqrt[3]{x}$, and you are tasked with identifying the correct equation that represents $g(x)$, which is the translated version.

Step-by-step approach:

1. Original Function: The function $y = \sqrt[3]{x}$ is symmetric around the origin and passes through the origin $(0, 0)$. Any transformation (shift up, down, left, or right) will modify its position.

2. Vertical Translation: In the graph, the function appears to have been shifted vertically compared to $y = \sqrt[3]{x}$. It crosses the $y$-axis above the origin (specifically at $y = 1.5$).

3. Translation of $y = \sqrt[3]{x}$:

   • A vertical shift upward by $1.5$ units can be described by the equation: $g(x) = \sqrt[3]{x} + 1.5$

Thus, the correct equation that represents $g(x)$ is: $g(x) = \sqrt[3]{x} + 1.5$

This corresponds to the third option in the list.

Summary:

The correct answer is:
$\boxed{g(x) = \sqrt[3]{x} + 1.5}$

Let me know if you want further details or have any other questions! Here are some related questions to consider:

1. How would the graph change if the function was translated horizontally instead of vertically?
2. What happens if we translate the original function by both a horizontal and vertical shift?
3. Can you explain the effect of translating a cubic root function on the graph's symmetry?
4. How would the equation change if the shift was downward instead of upward?
5. What would happen to the graph if we reflected $\sqrt[3]{x}$ over the x-axis?

Tip: When identifying translations, always look at key points like where the graph crosses the axes to determine the nature of the shift.