Final answer:
The composition of two one-to-one functions, a and b, is also one-to-one. This is proven by showing that if a∘b(x) equals a∘b(y), then x must equal y due to the injective nature of both functions a and b separately.
Step-by-step explanation:
To determine if the composition of two functions, a and b, is one-to-one, we must understand the definition of a one-to-one (injective) function. A function f is one-to-one if it assigns a unique output to every distinct input, meaning if f(x) = f(y), then x = y. For the composition a∘b to be one-to-one, whenever a∘b(x) = a∘b(y), it must follow that x = y.
To demonstrate this, let's assume that a∘b(x) = a∘b(y). Since b is one-to-one, the outputs of b from inputs x and y must be distinct if x and y are distinct, therefore b(x) = b(y) implies x = y. Next, because a is also one-to-one, the input into a that yields a∘b(x) must be unique; hence, if a(b(x)) = a(b(y)), then b(x) = b(y), and since b is one-to-one, x = y. This chain of logic confirms that a∘b is indeed one-to-one. Therefore, among the available choices, the correct answer is B) a∘b is one-to-one.