Question

Which of the following functions is a one-to-one correspondence?

A f: ℤ → ℤ, where f(n) = |n - 3|
B f: ℝ → ℝ, where f(x) = x³ + 1
C f: ℤ → ℤ, where f(n) = ⌈n/2⌉ + 2
D f: ℝ → ℝ, where f(x) = 2x² + 1

Answer 1

Option B, f(x) = x³ + 1, is the one-to-one correspondence among the choices because a cubic function is injective over the real numbers, meaning no two different inputs produce the same output.

Step-by-step explanation:

The question asks which of the given functions is a one-to-one correspondence. A one-to-one correspondence, also known as an injective function, is a function where each element of the domain maps to a unique element in the codomain, meaning that no two different inputs yield the same output.

Looking at the options:

• Option A, the absolute function f(n) = |n - 3|, is not one-to-one because, for example, f(2) and f(4) will both produce the same value, 1.
• Option B, the cubic function f(x) = x³ + 1, is one-to-one because for any real numbers x and y, if x ≠ y, then x³ ≠ y³, so f(x) ≠ f(y).
• Option C, the ceiling function f(n) = ⌈n/2⌉ + 2, is not guaranteed to be one-to-one as two different integers could round up to the same integer upon division by 2.
• Option D, the quadratic function f(x) = 2x² + 1, is not one-to-one because it produces the same output value for each pair of inputs that are negatives of each other, for instance, f(1) = f(-1).

Thus, the one-to-one function here is Option B, f(x) = x³ + 1.