What is function?
A function is a rule or formula that binds each element in set X (the definition) to exactly one and only one element in set Y (image). Not every element in Y has to be an output, and different elements in X can have the same output. A rule that assigns each element x in X to exactly one element y in Y is called a function f from a set of X elements to a set of Y elements. A function can also be thought of as a collection of ordered pairs (x, y). For every y-value in the set, there is only one x-value "but different x's can give the same y". This indicates that no x value with different y values can appear more than once. To define a function in this context, we typically use the rule y = f(x) and then plot the ordered pairs (x, f(x)) in the Cartesian Plane to produce a graph of that function. Using this graphical representation, we can use a test to determine whether we have a graph of a function: The Vertical Line Test is a method for determining whether an equation is a graph. But before that, it will be useful to look at the basic properties of functions.
Dimension set of function
For a function f(x)=y, let the domain of f be the set X. When we express a function as a collection of ordered pairs (x, y), these x values correspond to the domain. If only the rule y = f(x) is given, the domain is taken as the set of all real x's for which the function is defined. Value Set of the Function f: The value set of the function X -> Y is the set of y values that provide y = f(x) for some x in X. When we express a function as a collection of ordered pairs (x, y), these y values correspond to the set of values.
Graphical Behavior of Functions
As part of our investigation of how functions change, we can identify intervals where the function changes in certain ways. If function values get larger as input values increase within a range, we say that the function increases in that range. Similarly, if function values decrease as input values increase across an interval, the function decreases over that interval.
Vertical Line Test
According to the Vertical Line Test, if you cannot draw a vertical line that intersects a graph at more than one point (green), the graph is a function. Otherwise it is not a function (red):
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