Final answer:
Not all one-to-one functions are always increasing or decreasing; their behavior depends on the specific function in question. The correct answer is (c) Depends on the Function.
Step-by-step explanation:
Are one-to-one functions always increasing or decreasing? This is a common question about the nature of one-to-one functions in mathematics. The correct option to this query is (c) Depends on the Function. A one-to-one function, also known as an injective function, is a function where each input corresponds to a unique output. This does not demand that the function always be increasing or decreasing.
An example of an increasing one-to-one function is f(x) = x, where for every increase in x, there is a corresponding increase in f(x). Conversely, a decreasing one-to-one function could be f(x) = -x, where for every increase in x, there is a corresponding decrease in f(x). However, one-to-one functions may also be neither globally increasing nor decreasing, showing a combination of increasing and decreasing intervals. An illustration of this type is f(x) = x^3, which decreases for negative values of x and increases for positive values of x, while still maintaining a one-to-one correspondence between x and f(x).
Therefore, the behavior of a one-to-one function concerning increasing or decreasing trends is reliant on the particular function being considered. Hence, we choose only one option, (c) Depends on the Function, as the mentioned correct option in the final answer.