Let f be the squaring function defined in Example. Find f (–1), f (0), and .ExampleFunctio...

Let f be the squaring function defined in Example. Find f (–1), f (0), and .

Example

Functions Defined by Formulas

The squaring function f from R to R is defined by the formula f (x) = x2 for all real numbers x. This means that no matter what real number input is substituted for x, the output of f will be the square of that number. This idea can be represented by writing f (□) = □2. In other words, f sends each real number x to x2, or, symbolically, f : x → x2. Note that the variable x is a dummy variable; any other symbol could replace it, as long as the replacement is made everywhere the x appears.

The successor function g from Z to Z is defined by the formula g(n) = n + 1. Thus, no matter what integer is substituted for n, the output of g will be that number plus one: g(□) = □ + 1. In other words, g sends each integer n to n + 1, or, symbolically, g: n → n + 1.

An example of a constant function is the function h from Q to Z defined by the formula h(r ) = 2 for all rational numbers r . This function sends each rational number r to 2. In other words, no matter what the input, the output is always 2: h(□) = 2 or h: r → 2.

The functions f, g, and h are represented by the function machines in Figure.

Figure

A function is an entity in its own right. It can be thought of as a certain relationship between sets or as an input/output machine that operates according to a certain rule. This is the reason why a function is generally denoted by a single symbol or string of symbols, such as f, G, of log, or sin.

A relation is a subset of a Cartesian product and a function is a special kind of relation. Specifically, if f and g are functions from a set A to a set B, then

f = {(x, y) ∈ A × B | y = f (x)} and g = {(x, y) ∈ A × B | y = g(x)}.

It follows that

f equals g, written f = g, if, and only if, f (x) = g(x) for all x in A.