The Chemistry Maths Book, Second Edition

32 Chapter 2Algebraic functions

EXAMPLE 2.2Replace the variable xin (2.2) by the variable a.

f(a) 1 = 12 a

2

1 − 13 a 1 + 11

0 Exercise 4

EXAMPLE 2.3Replace the variable xin (2.2) by the functionh 1 + 12.

f(h 1 + 1 2) 1 = 1 2(h 1 + 1 2)

2

1 − 1 3(h 1 + 1 2) 1 + 11

= 1 2(h

2

1 + 14 h 1 + 1 4) 1 − 1 3(h 1 + 1 2) 1 + 11

= 12 h

2

1 + 18 h 1 + 181 − 13 h 1 − 161 + 11

= 12 h

2

1 + 15 h 1 + 131

= 1 g(h)

where

g(x) 1 = 12 x

2

1 + 15 x 1 + 13

is a new function of xthat is related tof(x)byg(x) 1 = 1 f(x 1 + 1 2).

0 Exercises 5, 6

EXAMPLE 2.4Replace the variable xin (2.2) by the differential operator (see

Chapter 4).

is a new differential operator.

EXAMPLE 2.5By the equation of state of the ideal gas, the volume is a function of

pressure, temperature, and amount of substance,

V 1 = 1 f(p,T,n) 1 = 1 nRT 2 p

and by the calculation performed in Example 1.1

f(10

5

Pa, 298 K, 0.1 mol) 1 = 1 2.478 1 × 110

− 3

m

3

2.2 Graphical representation of functions

A real function may be visualized either by tabulation or graphically by plotting.

Consider the function

y 1 = 1 f(x) 1 = 1 x

2

1 − 12 x 1 − 13 (2.3)

f

d

dx

d

dx

d

dx

d

dx













=













−













2312 += −

2

2

2

331

d

dx

• 
  

d

dx