Functions
A function is a special kind of relation. A relation is called a function if it has only one output for any given input. The relation R given as an example above is a function. A relation that is not a function is the square root relation:
Q: aQb if b 2 = a.
The domain A of Q is the non–negative numbers, and the range B of Q is all real numbers. Since every non–negative number has two square roots, one positive and one negative, Q is not a function. For example, 1Q1 and 1Q(–1), so the input 1 has two outputs, 1 and –1.
A simple idea of function is that a function is a set of ordered pairs where one quantity depends on another one to exist. This idea creates the function–notation f(x). For example, in f(x) = 3x 1, the expression 3x 1 depends on x to exist. As a consequence, the concept of independent variable for x and dependent variable for y is used. In this case, the function is written as y = 3x 1. In some cases, the independent variable x is also called argument, while the resultant y value is called image.
Another example of a function is the relation between people and their ages. At a given time, one person can only have one age; while several people could have the same age. This relation about ages, it is indeed a function. The following mapping diagrams illustrate this idea.
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