Denition and some Properties of Generalized Elementary Functions of a Real Variable I. Introduction The term elementary function is very often mentioned in many math classes and in books, e.g. Calculus books. In fact, the very vast majority of the functions that students come across are elementary functions of a real variable. However, there is a lack of a precise mathematical denition of elementary functions. Only a few authors in their textbooks, e.g. Stewart in his Calculus books try to give a description of elementary functions. Unfortunately, these descriptions are not given properly. For example, from the descriptions of elementary functions in Stewart's book, one could conclude that the function sin x, x 5 f(x) = ln x, x > 5 is elementary!! This is simply incorrect. Thus, this note is written to introduce a precise mathematical denition of generalized elementary functions of a real variable, which is a most broader class of functions that includes all the elementary functions. It is not claimed to be an original research article, but rather a note that could serve the students to see a proper mathematical denition of the term Generalized Elementary Function of a Real Variable. After the denition is introduced, it is easy to see that the generalized elementary functions of a real variable possess properties (which all elementary functions also possess) that could greatly simplify the mathematical analysis needed to be done on them. Also, many problems in mathematics deal with generalized elementary functions or even if the functions are non-elementary, very often the studying of these non-elementary functions lead to generalized elementary functions. According to the denition of generalized elementary functions given in this note, there are functions that are usually considered to be non-elementary, but are generalized elementary functions according to the denition. Since the problems are much simplied when the functions involved are generalized elementary functions, it is a good idea to have a precise mathematical denition of the generalized elementary functions. This denition and some more properties of generalized elementary functions are provided in this note. II. Denitions Denition. The following ve functions are referred to as the fundamental elementary functions of a real variable f (x) = c, c R with domain D R (D ) f 2 (x) = x, with domain D R (D ) f 3 (x) = e x, with domain D R (D )
f 4 (x) = sin x, with domain D R (D ) f 5 (x) =, with domain D R \ 0} (D ) x Denition 2. For any two functions (of a real variable) f(x) and g(x) with domains D f, D g and ranges R f, R g, respectively, the following operations are called The Fundamental Elementary Operations on Functions:. Addition: x D = D f Dg, f(x) and g(x) are both dened and have values a and b (a R and a R) respectively. Thus, x D there is a unique corresponding real number c = a + b. Hence, we dene a new function O (x) = f(x) + g(x) where the domain of O (x) is D = D f Dg, which is called the sum of f(x) and g(x). 2. Multiplication: In a similar fashion as in, we dene O 2 (x) = f(x) g(x) where the domain of O 2 (x) is D = D f Dg. O 2 (x) is called the product of f(x) and g(x). 3. Composition of Functions: x D f and R f D g, f(x) is dened and has a value f(x) = a, a R. Since x D f, a R f D g, hence a D g. Since a D g, g(a) = g(f(x)) is dened and has value g(a) = g(f(x)) = b. Thus, we dene O 3 (x) to be the composite function of f(x) and g(x) i R f D g O 3 (x) = g(f(x)), x D O3 (x) = D f 4. Inverse Functions: The function g(y) (domain D g and range R g ) is called inverse (function) of f(x) (domain D f and range R f ), i. R g D f 2. D g = R f 3. y D g, f(g(y)) = y We shall also call the operations that could be obtained from the above four fundamental operations an elementary operation. Denition 3. A function F (x) with domain D is called an invertible function if F (x) has a unique inverse function. Usually, this unique inverse function of F (x) is denoted as F (x). Denition 4. A function F (x) with domain D is called a generalized elementary function, if it can be obtained from one and the same set of fundamental elementary functions using a nite number of fundamental elementary operations in one and the same order. Remark. If F (x) is a generalized elementary function, then there exists exactly one formula to calculate the value of F (x) at any point x in its domain. Theorem. Subtraction could be obtained from applying the fundamental elementary operations, i.e. subtraction is an elementary operation. 2
Proof. f(x) g(x) = f(x) + ( g(x)) = f(x) + ( )(g(x)). Thus, the theorem follows. Theorem 2. Division could be obtained from applying the fundamental elementary operations, i.e. division is an elementary operation. Proof. Similarly done as in Theorem. Theorem 3. All polynomial functions are generalized elementary functions. Proof. A general polynomial function is dened as f(x) = a n x n + a n x n +... + a x + a 0 where n N, a n 0, a i R, i = 0,,..., n, and D f R. For an arbitrary n N, x n is obtained from n multiplication(s) of x and multiplication is a fundamental elementary operation. Thus x n is an elementary function. Since a i R (i = 0,, 2,..., n), a i is an elementary function. Since multiplication (a fundamental elementary operation) of two fundamental elementary functions is a generalized elementary function, the theorem is proven. Theorem 4. All rational functions are generalized elementary functions. Proof. A rational function could be dened as r(x) = f(x) g(x) where f(x) and g(x) are polynomial functions with domains D f and D g respectively. Then the domain of r(x) is D r = D f Dg \ x g(x) = 0}. The proof could be done in a similar fashion as in Theorem 3. Theorem 5. All algebraic functions are generalized elementary functions. Proof. Roots are inverses of powers (fundamental elementary operations) and rational powers could be dened in terms of roots, the theorem follows. Theorem 6. All trigonometric functions are generalized elementary functions. Proof. First, it will be shown that f(x) = cos(x), x R is a generalized elementary function. Note that cos x sin 2 x, cos x = ± sin 2 x depending on values of x. Thus, it is not a good idea to go this way. However, sin(x + π) = cos x. Since, x R is a fundamental elementary function and π is a fundamental 2 2 elementary function, x + π is an elementary function. Since sin x (x R) is a fundamental elementary 2 function, the composition of sin x with x + π which gives sin(x + π ) = cos x is also an elementary function 2 2 Since cos x is an elementary function, tan x = sin x cos x, cot x =, sec x =, and csc x = are all cos x sin x cos x sin x generalized elementary functions. Thus, all trigonometric functions are generalized elementary functions. It follows that all trigonometric function of generalized elementary functions are also generalized elementary functions. Theorem 7. ln x (x > 0) is a generalized elementary function. Proof. Since ln x (x > 0) is the inverse function of e x (fundamental elementary function), ln x (x > 0) is a generalized elementary function. Theorem 8. All logarithmic function of generalized elementary functions are generalized elementary functions. 3
Proof. log u(x) v(x) = ln v(x) ln u(x) = ln v(x) ln u(x) for values of x such that u(x), u(x) > 0, and v(x) > 0. Since ln v(x) is a generalized elementary function (composition of ln x and v(x)) and ln u(x) is also a generalized elementary function (composition of x, ln x and u(x)), ln v(x) ln u(x) = log u(x) v(x) is a generalized elementary function. Theorem 9. Functions of the form u(x) v(x) (u(x) > 0), where u(x) and v(x) are generalized elementary functions, are also generalized elementary functions.. Proof. For all values of x such that u(x) and v(x) are dened and greater than 0, u(x) v(x) = e v(x) ln u(x). Since the operations involved are either the fundamental elementary operations or a combination of them, u(x) v(x) is a generalized elementary function. Denition 5. For a function f(x) dened in a domain D, a point a is called an isolated point of D i. a D 2. ɛ > 0 such that x (a ɛ, a) (a, a + ɛ), x / D. The following statements are theorems that are very often proven in textbooks and thus will only be stated without proofs. All ve, f (x) - f 5 (x), fundamental elementary functions are continuous everywhere in their domains except at the isolated points and are discontinuous at the isolated points. 2. The sum of two continuous functions is also continuous everywhere in its domain except at the isolated points and is discontinuous at the isolated points. 3. The product of two continuous functions is also continuous everywhere in its domain except at the 4. The composition of two continuous functions is also continuous everywhere in its domain except at the 5. An inverse function of a continuous function is also continuous everywhere in its domain except at the Thus, one has the following very important theorem concerning generalized elementary functions Theorem 0. All generalized elementary functions are continuous in their domains, except at the isolated points at which they are discontinuous. Proof. This proof consists of applying various well-known theorems on continuity of functions. Theorem. The function is a generalized elementary function. x = x, x 0 x, x < 0 Proof. x is usually misunderstood as being a non-elementary function, since the function has two separate formulas. However, it is possible to write x with only one formula. That is x = x 2 which has only the elementary operations, square and square root. elementary function. Thus x for x R is a generalized Theorem 2. For a function f(x) dened as f(x) = g(x) h(x), x < a, x > a where g(x) is a generalized elementary function in D g = (, a) and h(x) is a generalized elementary function in D h = (a, + ). The function f(x) with domain D f = R \ a} is a generalized elementary function. 4
Proof. Note that and x + x 2x x + x 2x = = 0, x < 0, x > 0, x < 0 0, x > 0 Applying a small shifting in x, the function f(x) could be written as x a x a x a + x a f(x) = g(x) + h(x) 2(x a) 2(x a) with domain D f = R \ a}. Since the formula to calculate the value of f(x) at any point in the domain D f of f(x) consists of only generalized elementary functions and elementary operations, f(x) is a generalized elementary function. Note: The precise denition of the elementary functions (not generalized elementary functions) will be given in another note. I would like to thank my graduate student Tan Nguyen for his contribution to editing/typing of this text. Angel S. Muleshkov, Ph.D. Associate Professor of Mathematics, UNLV, 989 5