1.1: Real Functions
A mathematical function, denoted \(f\) , takes an input \(x\) (which is also called an argument ), and returns an output \(f(x)\) . For now, we consider the case where both \(x\) and \(f(x)\) are real numbers. The set of possible inputs is the function’s domain , and the set of possible outputs is the range .
Every function must have a well-defined output: for any \(x\) in the domain, \(f(x)\) must be a specific, unambiguous number. In other words, a function must be either a one-to-one (injective) mapping or a many-to-one mapping; the mapping cannot be one-to-many or many-to-many:
Simple examples of functions are those based on elementary algebra operations: \[ \begin{align} f(x) &= x + 2 \,\;\;\qquad\qquad \text{(a one-to-one function)} \\ f(x) &= x^2 + 2x + 4 \qquad \text{(a many-to-one function)}\end{align}\]