A mathematical function is a relationship between two sets of numbers, where each input from the first set (called the domain) is associated with exactly one output in the second set (called the range). Functions can be represented in various ways, including equations, graphs, and tables. For example, the function f(x) = 2x + 3 takes an input x and produces an output by multiplying x by 2 and then adding 3. Functions are essential in mathematics because they help describe how quantities change in relation to one another. They are used in various fields, including physics, economics, and engineering, to model real-world situations. Understanding functions allows us to analyze patterns and make predictions based on input values.