Final answer:
None of the provided pairs a), b), c), or d) demonstrate functions that are perfect inverses of each other, based on the properties of exponentials and logarithms.
Step-by-step explanation:
To identify which pair of functions are inverses of each other, we need to determine which pair 'undoes' each other. Recalling that the exponential function and its inverse, the logarithmic function, undo each other, we should look for a pair where one function is the exponential of a base and the other is the logarithm with the same base.
Let's analyze each provided pair:
- Pair a) is not inverse because while the first is a sum of logarithms, the second is not a corresponding exponential function.
- Pair b) looks promising because the logarithmic function is subtracting a constant, and the exponential function implies that constant could be a part of the exponent; however, the bases are not the same, and the operations do not precisely undo each other.
- Pair c) is not a proper inverse because the first function raises the logarithm to the fourth power, which does not directly correspond to the form of the second function which is a product of a number and an exponent.
- Pair d) shows two functions that are inverses of one another. The first function is a sum of logarithms, which is equivalent to the logarithm of the product due to the property log xy = log x + log y. So, this simplifies to log₅(7x). The second function is an exponential function 5ˣ + 57. However, on closer inspection, we can see that the +57 does not make the functions inverses of each other, because for two functions to be inverses, every operation in one function must be undone by the other and this additional 57 breaks this condition.
After reviewing all pairs, it appears that there is a typographical or conceptual error in the question, and none of the provided options show two functions that are perfect inverses of each other following the inverse operations of exponentials and logarithms.