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Understanding functions, in general, involves analyzing their definitions, domains, and codomains. A function takes an input, processes it, and gives an output. To determine if a given function is one-to-one, you need to ensure that each distinct input maps to a distinct output. This means if two different inputs ( a and b ) are provided to the function, they should yield different results.
For example, let's consider the function f(n) = n - 1 from the problem. If we assume f(a) = f(b) , then we get a - 1 = b - 1. Solving this equation, it becomes evident that a = b, confirming that the function is indeed one-to-one.
This type of analysis is crucial for understanding various types of functions and their properties. It helps identify whether changes in the input produce unique changes in output, which is key in functions considered in advanced mathematics.

An injective function is another term for a one-to-one function. For a function f to be injective, every element in its domain must map to a distinct element in the codomain. In formal terms, if f(a) = f(b) , then a = b.
Take the function f(n) = n^3 from the given exercise. If we assume f(a) = f(b) , then a^3 = b^3. This equation simplifies to a = b, showing that f(n) = n^3 is injective because each input has a unique output.
Conversely, not all functions are injective. For example, consider f(n) = n^2 + 1 . If f(a) = f(b), then a^2 + 1 = b^2 + 1 , which simplifies to a^2 = b^2. This equation has two solutions, a = b or a = -b, meaning different inputs can produce the same output. Therefore, f(n) = n^2 + 1 is not injective.

Integer functions are functions where the domain and codomain consist solely of integers. These functions are particularly useful in discrete mathematics. In the problem presented, all functions map integers to integers, making each function an integer function.
Consider the function f(n)=[n / 2]. This notation represents the floor function that maps any real number to the greatest integer less than or equal to that number. For integers n, this can mean f(2) = f(3) = 1. Since different input values can result in the same output, f(n)=[n / 2] is not a one-to-one function.
Integer functions often have specific properties. For instance, the function f(n) = n-1 decreases the input by 1, while f(n) = n^3 maps each integer to its cube. Verifying these properties helps understand their behavior and determine if they are injective.