Chapter 2 — Functions and Dynamic Relations¶
2.2 — Explicit and Implicit Domain: the Structural Restrictions¶
Before asking what a function does, one must ask where it can do it.
The definition of §2.1 requires a function \(f : A \to B\) to declare its domain and its codomain as part of the function itself. In practice, however, a real function is usually presented only by its formula, without mentioning either \(A\) or \(B\), and from now on that will be the norm in this book, unless stated otherwise. So that this is not an omission but a precise convention, we establish the following. When a function is given only by its formula, its codomain is \(\mathbb{R}\), and its domain is the largest subset of \(\mathbb{R}\) on which the formula produces a well-defined real number. That domain is called implicit. It is opposed to the explicit domain, which is the one declared expressly, for instance when a problem restricts \(x\) to an interval for reasons unrelated to the formula, such as a time that cannot be negative. This section deals almost entirely with the former.
§1. The structural restrictions of \(\mathbb{R}\)¶
Computing an implicit domain is not an open-ended task. It consists in identifying, within the formula, the operations that are not defined for every real and excluding the values of \(x\) that would leave them meaningless. With the operations defined so far, those restrictions are few, and all of them come from §1.1 and §1.2.
Denominators that vanish. Division by zero is not defined, and this comes from the axioms of §1.1: zero has no multiplicative inverse. If \(f(x)\) contains a quotient \(\frac{P(x)}{Q(x)}\), every value of \(x\) that makes \(Q(x)\) vanish is excluded from the domain. It is the same condition that governs the domain of a rational expression in §1.4; the only new thing is the name of the whole object, which is now a function. Powers with negative integer exponent fall under this case, because \(x^{-n} = \frac{1}{x^n}\) is a quotient: they require \(x \neq 0\).
Even-index roots of negative numbers. By §1.2, if \(n\) is even, \(\sqrt[n]{a}\) is defined in \(\mathbb{R}\) only for \(a \ge 0\). If \(f(x)\) contains \(\sqrt[n]{g(x)}\) with \(n\) even, the domain requires \(g(x) \ge 0\), a condition generally solved with the inequalities of §1.7. Odd-index roots, on the other hand, are defined for every real.
Fractional exponents. In §1.2, the power \(a^{m/n}\) with a fractional exponent was defined only for bases \(a > 0\), because with negative bases the value would depend on how the fraction is written. Therefore an expression such as \(x^{1/3}\) requires \(x > 0\), even though \(\sqrt[3]{x}\) is defined for every \(x\). Likewise, \(a^0 = 1\) was defined for \(a \neq 0\), and \(0^0\) was left undefined.
The other available operations, addition, subtraction, multiplication and powers with natural exponent, are defined for every real and impose no restrictions. Computing an implicit domain is, then, locating these patterns within the formula and translating each one into a condition on \(x\).
Example. Let \(f(x) = \dfrac{\sqrt{x + 3}}{x - 1}\). There are two restrictions. The radical requires \(x + 3 \ge 0\), that is, \(x \ge -3\). The quotient requires \(x - 1 \neq 0\), that is, \(x \neq 1\). Since both operations appear in the same formula, both conditions must hold at once, and the domain is the intersection of the sets each one determines:
with the interval notation of §1.1 and the symbol \(\setminus\) of §1.4, which removes from a set the elements of another.
Reminder. With the operations available so far, the implicit domain excludes the values that make a denominator vanish, those that make an even-index radicand negative, and those that fail the positive-base condition of a fractional exponent. When there are several restrictions, the domain is the intersection of the conditions, never the union.
§2. The scope of this list¶
It is worth saying explicitly something the list above does not say, so as not to suggest it by omission: it is not the list of all the restrictions a formula may ever impose. It is the complete list of those arising from the operations defined up to this point in the book. The catalog of functions will be enlarged later on, with logarithms and inverse trigonometric functions, among others, and each enlargement will bring its own condition of existence, as structural for that function as division by zero is for a quotient. The implicit domain is not, then, a fixed property of "the real function" in the abstract, but a consequence of which operations are defined at each moment. This section fixes the state of the list today; when the book's vocabulary grows, the list will be enlarged, but it will not need to be corrected.
§3. Restriction and equality of functions¶
With the definition of §2.1 we can specify when two functions, perhaps presented differently, are the same. A function is determined by its domain, its codomain and its set of pairs, and two functions are equal when they agree in all three.
Definition. Two functions \(f : A \to B\) and \(g : C \to D\) are equal, and we write \(f = g\), if and only if
With the convention of this section, real functions given by formulas all have codomain \(\mathbb{R}\), so the second condition always holds and the comparison reduces to the domains and the values. Those two conditions are independent: two formulas that look like the same rule can define different functions if their domains differ, and this is the most common error when comparing functions.
To describe the relation between two such functions, one more concept is useful. If \(f : A \to B\) and \(S \subseteq A\), the restriction of \(f\) to \(S\) is the function \(f|_S : S \to B\) defined by \(f|_S(x) = f(x)\) for all \(x \in S\). The vertical bar with the subscript indicates that the rule of \(f\) is kept but applied only to the elements of \(S\).
Example. Let \(f(x) = x\) and \(g(x) = \dfrac{x^2}{x}\). It is tempting to declare \(f = g\), because they look like the same rule. But \(g\) contains a quotient, and by §1, \(\operatorname{Dom}(g) = \mathbb{R} \setminus \{0\}\), whereas \(\operatorname{Dom}(f) = \mathbb{R}\). The simplification \(\frac{x^2}{x} = x\) holds, by §1.4, only where the original expression is defined, and the simplified expression does not keep the restriction \(x \neq 0\). Therefore \(f \neq g\): they are different functions, even though they agree at every point except \(x = 0\). The correct statement is \(g = f|_{\mathbb{R} \setminus \{0\}}\): \(g\) is the restriction of \(f\) to the nonzero reals.
Reminder. Two real functions given by formulas are equal if they have the same domain and the same rule on all of that domain. Agreeing in the formula after simplifying is not enough if the simplification altered the domain.
The domain is not a formality prior to what matters about a function, just as §1.4 warned that it was not one for a rational expression. It is part of the very identity of the object: changing the domain, even with the same formula, produces a different function. That is why determining the implicit domain is the first mathematical act any function demands, and not an administrative step dispatched before getting started.
With the domain fixed as a frontier of existence and as part of the identity of a function, it is time to examine its visual counterpart: which properties of a function, such as its zeros, its sign, its symmetries or its growth, can be read directly from its graph. That is the topic of the next section.