https://file.aiquickdraw.com/mathbot/file/1726447138353u4b1e6ls.jpg

The image represents a piecewise function \( f(t) \), which is defined differently depending on the value of \( t \).

### Function Breakdown:
- For \( 0 \leq t \leq 8 \):
  \[
  f(t) = \frac{5 \cdot \sqrt[3]{t}}{10} = 0.5 \cdot \sqrt[3]{t}
  \]
  This means the function is equal to half the cube root of \( t \), for values of \( t \) between 0 and 8 inclusive.

- For \( t > 8 \):
  There is no specified function for this interval in the image, so we assume that the function is undefined or constant for \( t > 8 \). More information would be needed to complete this part.

### Interpretation:
- The function behaves differently based on whether \( t \) is within the range \( [0, 8] \) or exceeds \( 8 \).
- For \( 0 \leq t \leq 8 \), the function is a scaled cube root function.

Would you like more details or have further questions?

Here are five related questions to expand your understanding:
1. What is the value of \( f(t) \) at \( t = 0 \) and \( t = 8 \)?
2. How would you graph this piecewise function for \( t \in [0, 8] \)?
3. What is the behavior of \( f(t) \) as \( t \) approaches 0?
4. Can we define a specific value for \( f(t) \) when \( t > 8 \)?
5. How would you modify this function to be continuous for all \( t \)?

**Tip:** When working with piecewise functions, always check for continuity and smooth transitions between different parts of the function.

Explore the piecewise function defined by f(t) = 0.5 * cube root of t for values of t between 0 and 8. This guide explains key concepts such as function domains, cube roots, and how piecewise functions work. Suitable for high school students in Grades 10-12.

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