Determine Surjectivity of Linear Function

Determine if the function $f(x) = 5x + 2$ is surjective from the integers to the integers.

When determining whether a function is surjective, also known as "onto," we need to explore if every possible output in the codomain is achieved by some input from the domain. For this problem, the function in question is linear, which simplifies certain aspects of the evaluation. Linear functions of the form f(x) = mx + b, where m and b are constants, are important foundational concepts in mathematics. Understanding how these functions behave provides insights into more complex topics in calculus and linear algebra.

For a function to be surjective from the set of integers to itself, every integer must be attainable as an output from plugging in some integer into the function. This typically involves investigating whether the inverse operation of the function is well-defined over the codomain. In simpler terms, we verify if there is an integer input for every integer output possible under f(x). Since the function given here, $f(x) = 5x + 2$, involves multiplication and addition, we delve into properties of integers related to divisibility and completeness of integer coverage.

The problem serves as an excellent introduction to the application of elementary logic and algebra to verify mapping properties such as injectivity, surjectivity, and bijectivity. These properties are pivotal when working with functions in advanced mathematics. Through the exploration of these concepts, students learn to understand and categorize functions, an essential skill in the landscape of higher-level math.