Introduction to Game Programming with C++ (Wordware Game Developers Library)

9.1 Functions in Mathematics

In mathematics, when one quantity depends on a second quantity, the first is said to be a function of the second. The calculation of the area of a circle, for example, depends on the radius. So, the area of a circle is a function of the radius. Consider the following equation:

• y = 3x − 5

This equation can be considered a function in which x is the independent variable and y is the dependent variable. In terms of a function, x is the domain of the function, and can be any number. The entire scope of numbers x can assume is called the range, such as 1, 2, 3, and so on. y is said to be the value of the function. This makes sense because, for whatever number x can be, there will also be a corresponding value for y.

9.1.1 Function Notation

In mathematics there is a notation reserved for representing functions. Consider the following two functions:

These are two separate functions, and each can be given a name. Since f is often used to denote a function, we'll call the first one f and the second one g.

Here are the two functions f and g again. The parentheses are used after the name of the function and inside the parentheses are the arguments of the function.

These functions have one argument each, namely x. So, each function expects a value for x as its input. Consider the following:

f(x) = 5x + 3

Where x = 3:

f(3) = 15 + 3 = 18

Since functions evaluate to a single value, functions can also be used as arguments to other functions. This is demonstrated in the following example. g is evaluated first and its value is passed on to f as an argument.

• f(g(x))