Download De nition and some Properties of Generalized Elementary Functions

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 1.
The following ve functions are referred to as the fundamental elementary functions of a real
variable
f1 (x) = c, c ∈ R with domain D ⊆ R (D 6= ∅)
f2 (x) = x, with domain D ⊆ R (D 6= ∅)
f3 (x) = ex , with domain D ⊆ R (D 6= ∅)
1
f4 (x) = sin x, with domain D ⊆ R (D 6= ∅)
f5 (x) =
Denition 2.
Rg ,
1
, with domain D ⊆ R \ {0} (D 6= ∅)
x
For any two functions (of a real variable)
f (x)
and
g(x)
with domains
Df , D g
and ranges
Rf ,
respectively, the following operations are called The Fundamental Elementary Operations on Functions:
T
∀x ∈ D = Df 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
1. Addition:
a new function
O1 (x) = f (x) + g(x)
where the domain of
O1 (x)
is
D = Df
T
Dg ,
which is called the sum of
f (x)
and
g(x).
2. Multiplication: In a similar fashion as in 1, we dene
O2 (x) = f (x) · g(x)
D = Df Dg . O2 (x) is called the product of f (x) and g(x).
3. Composition of Functions: ∀x ∈ Df and Rf ⊆ Dg , f (x) is dened and has a value f (x) = a, a ∈ R.
Since x ∈ Df , a ∈ Rf ⊆ Dg , hence a ∈ Dg . Since a ∈ Dg , g(a) = g(f (x)) is dened and has value
g(a) = g(f (x)) = b. Thus, we dene O3 (x) to be the composite function of f (x) and g(x) i
(
Rf
⊆ Dg
O3 (x) = g(f (x)), x ∈ DO3 (x) = Df
where the domain of
O2 (x)
T
is
4. Inverse Functions: The function
(domain
Df
and range
Rf ),
g(y)
(domain
Dg
and range
Rg )
is called inverse (function) of
f (x)
i


1. Rg ⊆ Df
2. Dg = Rf


3. ∀y ∈ Dg , 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
Denition 4.
A function
F (x)
with domain
D
is called an invertible function if
−1
function. Usually, this unique inverse function of F (x) is denoted as F
(x).
F (x)
with domain
D
F (x)
has a unique inverse
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.
value of
If F (x) is a generalized elementary
F (x) at any point x in its domain.
function, then there exists exactly one formula to calculate the
Theorem 1. Subtraction could be obtained from applying the fundamental elementary operations, i.e. subtraction is an elementary operation.
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Proof. f (x) − g(x) = f (x) + (−g(x)) = f (x) + (−1)(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 1.
Theorem 3. All polynomial functions are generalized elementary functions.
Proof.
A general polynomial function is dened as
f (x) = an xn + an−1 xn−1 + ... + a1 x + a0
n ∈ N , an 6= 0, ai ∈ R, i = 0, 1, ..., n, and Df ⊆ R. For an arbitrary n ∈ N, xn is obtained from n
n
multiplication(s) of x and multiplication is a fundamental elementary operation. Thus x is an elementary
function. Since ai ∈ R (i = 0, 1, 2, ..., n), ai is an elementary function. Since multiplication (a fundamental
where
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)
f (x) and T
g(x) are polynomial functions with domains Df
r(x) is Dr = Df Dg \ {x|g(x) = 0}. The proof could be done in
where
and
Dg
respectively. Then the domain of
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,
p
cos x 6=
p f (x) = cos(x), x ∈ R is a generalized elementary
1 − sin x, cos x = ± 1 − sin2 x depending on values of x. Thus, it is not a
it will be shown that
2
function.
Note that
good idea to go this
way.
sin(x + π2 ) = cos x. Since, x ∈ R is a fundamental elementary function and π2 is a fundamental
π
elementary function, x +
is an elementary function. Since sin x (x ∈ R) is a fundamental elementary
2
π
π
function, the composition of sin x with x +
which gives sin(x + ) = cos x is also an elementary function
2
2
sin x
cos x
1
1
Since cos x is an elementary function, tan x =
, cot x =
, sec x =
, and csc x =
are all
cos x
sin x
cos x
sin x
However,
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
ex
(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.
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Proof. logu(x) v(x) =
ln v(x)
ln u(x)
= ln v(x) ·
1
for values of
ln u(x)
u(x) 6= 1, u(x) > 0, and v(x) > 0. Since
1
ln v(x) is a generalized elementary function (composition of ln x and v(x)) and ln u(x)
is also a generalized
1
1
elementary function (composition of , ln x and u(x)), ln v(x) ·
= logu(x) v(x) is a generalized elementary
x
ln u(x)
x
such that
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
the operations involved are either the fundamental elementary operations or a
0, u(x)v(x) = ev(x)·ln u(x) . Since
v(x)
combination of them, u(x)
is a generalized elementary function.
Denition 5.
1.
2.
a∈D
∃ > 0
For a function
such that
f (x)
dened in a domain
D,
a point
a
is called an isolated point of
D
i
S
∀x ∈ (a − , a) (a, a + ), x ∈
/ D.
The following statements are theorems that are very often proven in textbooks and thus will only be stated
without proofs
1. All ve,
f1 (x)
-
f5 (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
isolated points and is discontinuous at the isolated points.
4. The composition of two continuous functions is also continuous everywhere in its domain except at the
isolated points and is discontinuous at the isolated points.
5. An inverse function of a continuous function is also continuous everywhere in its domain except at the
isolated points and is discontinuous at the isolated points.
Thus, one has the following very important theorem concerning generalized elementary functions
Theorem 10. 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 11. The function
(
x
|x| =
−x
,x ≥ 0
,x < 0
is a generalized elementary function.
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| =
√
x2
which has only the elementary operations, square and square root.
Thus
|x|
for
x ∈ R
is a generalized
elementary function.
Theorem 12. For a function f (x) dened as
(
g(x) , x < a
f (x) =
h(x) , x > a
where g(x) is a generalized elementary function in Dg = (−∞, a) and h(x) is a generalized elementary function
in Dh = (a, +∞). The function f (x) with domain Df = R \ {a} is a generalized elementary function.
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Proof.
Note that
|x| + x
=
2x
(
0 ,x < 0
1 ,x > 0
and
−|x| + x
=
2x
Applying a small shifting in
x,
the function
f (x) = g(x)
with domain
Df
of
f (x)
Df = R \ {a}.
f (x)
(
1 ,x < 0
0 ,x > 0
could be written as
x − a − |x − a|
x − a + |x − a|
+ h(x)
2(x − a)
2(x − a)
Since the formula to calculate the value of
f (x)
at any point in the domain
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, 1989
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