Real Valued Functions — Domain and Range

A function $f : A \to B$ is called a real valued function if its co-domain $B$ is a subset of $\mathbb{R}$ (usually $B = \mathbb{R}$ itself), so that every output $f(a)$ is a real number. If, in addition, the domain $A$ is also a subset of $\mathbb{R}$, $f$ is called a real function of a real variable, or simply a real function. Every standard function studied later in this chapter — constant, identity, polynomial, rational, modulus, exponential, logarithmic, signum, greatest integer — is a real function in this sense, and each is normally given by a single algebraic formula, $y = f(x)$, rather than by listing ordered pairs.

Domain when it is not stated. When a real function is given only by its formula, with no domain mentioned, the domain is understood to be the largest subset of $\mathbb{R}$ for which the formula produces a well-defined real number. Finding this "natural domain" means watching for the operations that can fail:

• A denominator must never be zero, so any $x$ that makes the denominator vanish is excluded.
• The expression under an (even-order) square root must never be negative, so it must be $\ge 0$.
• The argument of a logarithm must be strictly positive. Once these exclusions are removed from $\mathbb{R}$, whatever remains is the domain.

Finding the range. The range of a real function is the actual set of values $f(x)$ takes as $x$ runs over the domain. Unlike the domain, there is no single universal recipe for the range — it is found by reasoning about the specific formula: solving $y = f(x)$ for $x$ in terms of $y$ and asking for which $y$ that gives a real $x$ back in the domain, or by directly tracking the largest and smallest values (and everything continuously in between) that the formula can output as $x$ sweeps across the domain. …