A real-valued function is a mathematical function that assigns a real number to each element in its domain. The domain can consist of real numbers, intervals, or sets of real numbers. For example, the function f(x) = x² takes any real number x and returns its square, which is also a real number. These functions can be represented graphically on a coordinate plane, where the x-axis represents the input values and the y-axis represents the output values. Common examples of real-valued functions include linear functions, polynomial functions, and trigonometric functions like sine and cosine.