Final answer:
Exponential functions are one-to-one with unique inputs and outputs, and this is proved by the fact that exponential functions and their inverses (natural logarithms) undo each other.
Step-by-step explanation:
Exponential functions are indeed one-to-one. To be defined as a one-to-one function, each input must be associated with exactly one output and each output must come from exactly one input. Because an exponential function plateaus out but never decreases or assumes the same value twice, it fits this definition. More formally, if f(x) is an exponential function, and f(a) = f(b), then a must equal b.
Moreover, the natural logarithm is the inverse of the exponential function, which further confirms their one-to-one property. This can be expressed with two important identities: ln(ex) = x and eln(x) = x. These pairs of functions 'undo' each other, a characteristic trait of inverse functions in one-to-one relationships. Therefore, the correct answer to the question 'Are exponential functions one-to-one?' is (a) Yes.