Let \({\mathbb{F}}\) be either the real field ℝ or the complex field ℂ. Recall that a scalar-valued function f : X → \({\mathbb{F}}\) on a nonempty set X is called real-valued if \({\mathbb{F}}\) = ℝ and complex-valued if \({\mathbb{F}}\) = ℂ. If it is not relevant whether a scalar-valued function is real-valued or complex-valued, then we also refer to it as \({\mathbb{F}}\) -valued. The kernel N(f) and range R(f) of a function f : X → \({\mathbb{F}}\) are the sets