The Chemistry Maths Book, Second Edition

2.6 Rational functions 51

(2.28) is defined for all values of xwith the exception of the roots of the polynomial

Q(x)in the denominator, for whichQ(x) 1 = 10.

The graph of the functiony 1 = 112 xin Figure 2.13 demonstrates some typical

properties of rational functions. As xapproaches zero from the right(x 1 > 1 0)the value

of 12 xbecomes arbitrarily large; we say thaty 1 = 112 xtends to

infinity as xtends to zero. Similarly, ytends to minus infinity

as xtends to zero from the negative side. The pointx 1 = 10

is called a point of singularity, and all rational functions

have at least one such point, one for each root ofQ(x). The

graph also shows that asx 1 → 1 ∞from either side, the curve

approaches the y-axis arbitrarily closely but does not cross

it. The y-axis is the linex 1 = 10 and is called an asymptote

to the curve; we say that the curve approaches the line

x 1 = 10 asymptotically. The liney 1 = 10 (the x-axis) is also an

asymptote.

Division of one polynomial by another

The function (2.28) is called a properrational function if the degree nof the

numeratorP(x) is smaller than the degreemof the denominatorQ(x), as in examples

(i) and (iv) of (2.29). Otherwise, as in examples (ii) and (iii), it is called improper.

In ordinary number theory, an improper fraction is one whose value is greater than

or equal to 1; for example 5 2 2or2 2 2. An improper fraction can always be reduced

to a combination of proper fractions by division. An improper rational function

is reduced to a combination of proper functions by algebraic division.

EXAMPLE 2.25Dividex

3

1 − 17 x

2

1 + 116 x 1 − 111 byx 1 − 11.

By adapting the method of ordinary long division, we write

It follows that

xx x

x

xx

x

32

2

71611

1

610

1

−+−

−

=−+−

−

xx

xx x

32

2

61611

−

−+−

subtract

into goes times

s

−−

−+

66

2

xx

xx uubtract

10 11 into goes times 10 10

10 1

xxx

x

−

− 00

1

subtract

− remainder

)

xx

xxx x x x x

2

32 3 2

610

1 7 16 11

−+

−−+− into goes tiimes

y

x

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Figure 2.13