Question

Determine whether each of these functions from z to z is one-to-one.

1) f(n) = n - 1
2) f(n) = n² + 1
3) f(n) = n³
4) f(n) = n/

Answer 1

Out of the given functions, f(n) = n - 1 and f(n) = n³ are one-to-one because they assign a unique output to each input. The function f(n) = n² + 1 is not one-to-one, as it can yield the same result for different inputs. The last function has insufficient information provided to determine its one-to-one status.

Step-by-step explanation:

In mathematics, when discussing functions, a function f from z to z is said to be one-to-one if it never assigns the same value to two different domain elements. In other words, for all values a and b in the domain, if a ≠ b, then f(a) ≠ f(b). Let's examine the given functions one by one to determine if they are one-to-one:

• f(n) = n - 1: Subtracting 1 from a number gives a unique result for each number, so different n's will produce different f(n)'s. Thus, this function is one-to-one.
• f(n) = n² + 1: Squaring a number and then adding one can result in the same output for different inputs, such as f(1) = f(-1) = 2. Hence, this function is not one-to-one.
• f(n) = n³: Cubing a number gives a unique result for each number. This function is one-to-one since different n's will produce different f(n)'s.
• f(n) = n: There seems to be some typo or information missing for this function. Ideally, the identity function, f(n) = n, is one-to-one.