👉 Mathematically, the concept of a function can be illustrated through its definition: A function \( f \) from a set \( A \) to a set \( B \), denoted as \( f: A \rightarrow B \), assigns to each element \( x \) in \( A \) exactly one element \( f(x) \) in \( B \). This is evident in the property that for every input \( x \) from \( A \), there exists a unique output \( f(x) \) in \( B \). Graphically, this translates to a one-to-one correspondence between points in \( A \) and \( B \), where no two elements in \( A \) map to the same element in \( B \). For example, if \( f(x) = x^2 \) maps \( A = \{1, -1\} \) to \( B = \{1\} \), it clearly shows the function's structure: each input has a unique output, and no two inputs map to the same output. This mathematical framework is foundational in algebra, calculus, and beyond, underpinning how we understand relationships between variables.
