To determine the transformation of the cube root parent function $f(x) = \sqrt[3]{x}$, we analyze the options provided. Each choice must represent a modification (stretch, shift, reflection, etc.) of the cube root function.

Step-by-step Analysis:

1. Cube Root Parent Function: $f(x) = \sqrt[3]{x}$ is the base function. Transformations typically involve:

   • Vertical/horizontal stretches or compressions.
   • Vertical/horizontal shifts.
   • Reflections across axes.

2. Options Analysis:

   • Option A: $g(x) = 34/2$: This is a constant, not a cube root transformation.
   • Option B: $g(x) = 32 + 3$: This is not a valid representation of a cube root function and appears unrelated.
   • Option C: $g(x) = 3\sqrt[3]{x}$: This matches a transformation of the cube root parent function, specifically a vertical stretch by a factor of 3.
   • Option D: $g(x) = \frac{1}{x} + 3$: This represents a reciprocal function, not related to the cube root function.

Conclusion:

The function $g(x) = 3\sqrt[3]{x}$ (Option C) is the correct transformation of the cube root parent function.

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Would you like further explanation or examples?
Here are 5 related questions to expand your understanding:

1. What are other transformations of $f(x) = \sqrt[3]{x}$ and how are they written?
2. How does a vertical stretch affect the graph of a function?
3. What is the difference between a vertical and horizontal transformation?
4. Can cube root functions have reflections, and how are they expressed?
5. How does $f(x) = \sqrt[3]{x} + k$ shift the graph?

Tip: Always compare the structure of transformations to the base function to identify shifts, stretches, or reflections easily.