Understanding Engineering Mathematics

You may think that the fraction

x

x

does exist, and that it is equal to 1, on cancellation of thex. But again this is only true
ifx =0. The expression is not defined forx=0, that is, there is no number equal to^00.
In general then,algebraic fractions are not defined for those values ofxwhich make
their denominator vanish.

Examples
1
x− 1

is not defined atx= 1

x− 2

x− 2

is not defined atx= 2

x+ 1

x^2 − 2 x− 3

is not defined atx=−1andx= 3 (why?)

Just as an arithmetic fraction such as^23 is called a rational number, a general algebraic
function of the form:

Polynomial

Polynomial

is called arational function. The polynomial on the top is thenumeratorthat on the
bottom thedenominatorin analogy with numerical fractions.

Examples
x+ 1
x+ 2

,

x^2 + 2 x− 3

x^2 + 4 x+ 2

are rational functions, whereas

√

x+ 1

x+ 2

,

x^2 + 3 x− 4

x+

√

x

are not because neither of

√

x+1orx+

√

xare polynomials.

Solution to review question 2.1.7

(i)

x− 1

x^2 + 1

and (iii)

x− 1

x^2 +x+ 1

are both of the form

polynomial

polynomial

and are therefore rational functions.

(ii)

√

x+

1

√

x

and (iv)

√

x− 1

x+ 1

are not rational functions because of the

square roots.