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To determine which of the given functions is a one-to-one function, we need to analyze the nature of each function:

1. **One-to-One Function**: A function is one-to-one if each input maps to a unique output. In other words, no two different inputs produce the same output. One way to test this is by using the **Horizontal Line Test** or checking the derivative to see if it is always positive or always negative.

### Analysis of Each Function
1. **\( b(x) = -0.52x^3 - 0.8x^2 + 3.1x - 3.8 \)**  
   This is a cubic function. Cubic functions can change direction (increasing and then decreasing), so they are generally not one-to-one.

2. **\( m(x) = \frac{16 - x^4}{4} \)**  
   This function includes a term with \( x^4 \), which is a polynomial of degree 4. Such functions are symmetric and not one-to-one.

3. **\( c(x) = -\frac{1}{3}x - 8 \)**  
   This is a linear function. Linear functions are always one-to-one because they have a constant rate of change and pass the Horizontal Line Test.

4. **\( h(x) = |x + 8| - 5 \)**  
   Absolute value functions are not one-to-one because they have a V shape and fail the Horizontal Line Test.

### Conclusion
The one-to-one function among the given choices is:

**\( c(x) = -\frac{1}{3}x - 8 \)**.

Would you like more details or have any other questions?

### Related Questions
1. How do you perform the Horizontal Line Test for different functions?
2. What are some common properties of one-to-one functions?
3. How does the derivative test help in identifying one-to-one functions?
4. What transformations can turn a non-one-to-one function into a one-to-one function?
5. How do piecewise functions affect the one-to-one property?

### Tip
Remember, a function is one-to-one if it never takes the same value twice. Checking for unique outputs can often be done visually or by using derivatives.

Learn how to determine which function is a one-to-one function using the Horizontal Line Test. This problem includes linear, cubic, and absolute value functions, making it suitable for students in Grades 9-12.

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