Page 111 - DMTH401_REAL ANALYSIS

Unit 9: Functions

Notes
Example: Prove that every rational function is an algebraic function.

Solution: Let f : R  R be given as
p(x)
f(x) = , " x R,
q(x)

where p(x) and q(x) are some polynomial functions such that q(x)  0 for any x  R.
Then we have

p(x)
y = f(x) =
q(x)
q(x) y– p(x) = 0
which shows that y = f(x) can be obtained by solving the equation

q(x) y – p(x) = 0.
Hence f(x) is an algebraic function.

A function which is not algebraic is called a Transcendental Function. Examples of elementary
transcendental functions are the trigonometric functions, the exponential functions and the
logarithmic functions, which we discuss in the next section.

9.2 Transcendental Functions

In earlier unit, we gave a brief introduction to the algebraic and transcendental numbers. Recall
that a number is said to be an algebraic if it is a root of an equation of the form

n–1
n
a x + a x + .... x + a x + a = 0
0 1 n–1 n
with integral coefficients and a  0, where n is a positive integer. A number which is not
0
algebraic is called a transcendental number. For example the numbers e and IT are transcendental
numbers. In fact, the set of transcendental numbers is uncountable. Based on the same analogy,
we have the transcendental functions. We have discussed algebraic functions. The functions that
are non-algebraic are called transcendental functions. In this section, we discuss some of these
functions.

9.2.1 Trigonometric Functions

You are quite familiar with the trigonometric functions from the study of Geometry and
Trigonometry. The study of Trigonometry is concerned with the measurement of the angles and
the ratio of the measures of the sides of a triangle. In Calculus, the trigonometric functions have
an importance much greater than simply their use in relating sides and angles of a triangle. Let
us review the definitions of the trigonometric functions sin x, cos x and some of their properties.
These functions form an important class of real functions.
2
2
Consider a circle x +y = r with radius r and centre at O. Let P be a point on the circumference
2
of this circle. If  is the radian measure of a central angle at the centre of the circle as shown in the
Figure 9.1 then you know that the lengths of the arc AP is given by
s = r.

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