Real-valued function

A real-valued function takes each function input to a real number. Its codomain can be all of $\mathbb{R}$ or a specified subset. Thus $f(x)=x^2$ on the real line is a real-valued function and has image $[0,\infty )$, whether the declared target is that interval or all of $\mathbb{R}$.

The inputs need not be real numbers: the cardinality of a finite set and the squared norm of a vector are also real-valued functions. The adjective describes outputs, while scalar-valued function allows outputs in other fields, notably the complex numbers.