The Chemistry Maths Book, Second Edition

2 Algebraic functions

2.1 Concepts

When the equation of state (1.1) of the ideal gas is written in the form

it is implied that the value of the volumeV(the dependent variable) is determined by

the values of the pressure p, temperatureT, and amount of substancen(the independent

variables). In general, a dependent variable is said to be a function of the variable or

variables on which it depends.

1

In this example,Vis a function of the three variables

p, T, and n. In the present chapter we are concerned with functions of only one

variable; the case of more than one independent variable is discussed in Chapter 9.

Let the variable ybe a function of the variable x. For example, in equation

y 1 = 12 x

2

1 − 13 x 1 + 11 (2.1)

the expression on the right of the equal sign defines the function f,

f(x) 1 = 12 x

2

1 − 13 x 1 + 11 (2.2)

whose value for any given value of xis to be assigned to the variable y(readf(x)as

‘f of x’). The function fis the rulefor calculating yfrom x.

A function has a numerical value when numerical values are assigned to the

variables.

EXAMPLE 2.1The values of the function (2.2) whenx 1 = 1 2,x 1 = 11 ,andx 1 = 10 are

f(2) 1 = 121 × 12

2

1 − 131 × 121 + 111 = 13

f(1) 1 = 121 × 11

2

1 − 131 × 111 + 111 = 10

f(0) 1 = 121 × 10

2

1 − 131 × 101 + 111 = 11

0 Exercises 1–3

The concept of function is more general than this however, because the variable xcan

be replaced by another variable, by a function, or by a more complicated quantity

such as a differential operator or a matrix.

V

nRT

p

=

1

The word ‘function’ was first used in this context by the German mathematician Gottfried Wilhelm Leibniz

(1646–1716).