Wednesday, October 12, 2016

What is the definition of a function?

As described above, a function is a rule that creates correspondence between an element of set x, called domain, and an element of set y, called range. The function is usually denoted as y = f(x), where f(x) means: if one takes x and applies the rule f, y is obtained.


Functions can be given in various forms:


1) Verbal description. For example, one's weekly salary depends on the number of hours worked per week,...

As described above, a function is a rule that creates correspondence between an element of set x, called domain, and an element of set y, called range. The function is usually denoted as y = f(x), where f(x) means: if one takes x and applies the rule f, y is obtained.


Functions can be given in various forms:


1) Verbal description. For example, one's weekly salary depends on the number of hours worked per week, if one makes $15 an hour.


2) Formula. For example, y = 2x + 3.


3) Table, or set of ordered pairs. For example, {(1, 2), (2, 3), (-1, 4)}. The first number in a pair is an element of x and the second number is an element of y.


4) Graph.


However, not every rule, or every correspondence between an element of set x and an element of set y is a function. In a function, each element of x can correspond to only one element of y. If this is not the case, the correspondence is called a relation, but it is not a function.


For example, the set of ordered pairs{(1, 2), (1, 3), (2, 5)} is NOT a function because for x = 1 there are two values of y: y = 2 and y = 3.


Likewise, the equation `y^2 = x^2` does NOT define a function y = f(x) because it is possible for x = 1 to have two corresponding elements of y: y = 1 and y = -1. On the graph, the curve representing this equation can be crossed by a vertical line in two places.


However, a function CAN have two different elements of x corresponding to one element of y. For example,


{(1, 2), (2, 2), (-1, 3)} is a function and `y = x^2` is a function.


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