Final answer:
To prove if a composition of functions is one-to-one, then the individual functions must also be one-to-one, one must understand what it means for a function to be one-to-one, and demonstrate that if the composition is injective, both functions must map distinct elements to distinct images.
Step-by-step explanation:
To prove that if a composition of functions is one-to-one, then the individual functions must also be one-to-one, we must establish what it means for a function to be one-to-one, or injective. A function f is said to be one-to-one if different elements in the domain map to different elements in the codomain. This can be shown if, for any two elements a and b in the domain, where a is not b, the function f(a) ≠ f(b). The idea that if the composition of two functions (g composed with f) is one-to-one, then the individual functions f and g must also be one-to-one, is a fundamental characteristic of injective functions.
To demonstrate this, assume we have two functions, f and g, such that their composition g(f(x)) is one-to-one. If we assume g is injective and we have two inputs x and y in the domain of f such that f(x) = f(y), then it follows that g(f(x)) = g(f(y)). However, since g(f(x)) is one-to-one, we must have x = y, which shows that f must be injective as well.