Final answer:
Without specific details of the functions mentioned, it is not possible to definitively determine if they are one-to-one. Indications of discontinuity, being double-valued, or divergence imply that a function is not one-to-one, as a function must be continuous, single-valued, and the outputs should not diverge to maintain the one-to-one mapping.
Step-by-step explanation:
To determine whether each of these functions is one-to-one, we need to establish if each function maps every element of the domain to a unique element of the codomain. For a function to be one-to-one, each value of the output z should be paired with exactly one input value x. The provided statements seem to involve variables and operations that need to be assessed for their ability to maintain a one-to-one correspondence between inputs and outputs.
Looking at the equations, the function 1(x) seems to indicate that for regions I and III, there's a general form that must be followed. It suggests that there's a condition tied to the regions which must be met. Without specifics about the function, it's not possible to conclude its one-to-one nature.
The mention of function 1(x) being discontinuous, double-valued, or divergent indicates a failure to be one-to-one. A one-to-one function must be continuous, single-valued, and the outputs should not diverge to maintain the one-to-one mapping.
In the mathematical operations where A, Z, and efn are discussed, verifying if they align with the principles of one-to-one functions would require further context. If the equations represent the behaviors of the functions across different inputs, the consistency of these outputs would be important for the functions to be one-to-one.
The concept of independence, as indicated with F and C being independent or not, can be analogous to discussing functions. If changing one variable doesn't affect the other, it is similar to a one-to-one function where each input has a unique output.