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.