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Home < More Algebra < 4 Functions << 7 Functions with finite domains
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Functions with finite domainsDomains and Ranges of functions y = f(x) may be defined as follows when y and x denote real numbers.
In the previous lessons, we saw how to give or describe a dependence of a number or quantity y on other numbers or quantities by a formula y = f(a,b,c). But we can also describe how dependence in other ways using forwards and backwards.
with arrow diagramsA function, dependency, map or assignment f may defined by arrow diagram.
The domain of f,
The range of f,
The range is a subset of the target set {a,b,c,d,6.} So the map is not surjective (onto) The map f is many to one as f(5) and f(2) are equal to c. Equivalent Ways with tablesWe could have defined the previous function with a horizontal table
or vertical table
as you like. Any letter may be used in place of x.
Another Table ExampleHere we use a table to define h(x).
The table says how to compute a function h. From the table, we may evaluate the mapping h at each element of its domain {0, 1, 2, 3, 4, 5, 6}
The domain of h is set of numbers
in the first row. The range of h is the set of numbers
in the second row. Yet another table exampleA table of values
in which there is no duplicate numbers or objects in the x-row gives a function f with
The range of f,
In this example
f(2) = 3 f(3) = -1 f(4) = 4 List MethodA function f may be described by specifying it values at points in a set. f(2) = 3, f(4) =-11 f(8) = 2 The foreging gives a function f with domain {3, 4, 8} and range {3, -11, 2} List Method in GeneralA function f defined for a set of distinct values x1, x2, ... xn. by specifying its values y1, y2, ... yn at those numbers, so that
Here the domain of definition of f,
is a finite set. The range of f
is a finite set. (Remember to eliminate duplicate values of y so that elements of the range are not listed twice.) Using ordered pairsA function f in mathematics may be specified by a set of ordered pairs. For example f = { (1,3.4), (2.5, 4), (2.1, 5), (-1, 8) } represents the function f with
The function domain, the set of items for which is defined, is
Plotting the ordered pairs gives the graph of f. The set of points
provides the graph of f. So we may write
and identify the function with its graph. The graph is a set of points in the coordinate plane. So the study of functions y =f(x) where y and x are real numbers becomes part of analytic geometry. The stage is now set for the following. Analytic Geometry View of Functions in the planeHere the set of points in the plane is denoted by IR2 = {(x,y) such x and y are real numbers} A finite set S of points (x, y) in the coordinate plane IR2 which satisfies the vertical line property, namely each vertical line intersect S at most one point. In this case, when the line x = a intersects the set S at a point (a,b), the computation associated rule f puts f(a) = b. Site to do: Put an illustration here ] The set S may be given by a list of order pairs or by their plot (graph) in the plane. |
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Home < More Algebra < 4 Functions << 7 Functions with finite domains
[1] [2] [3] [4] [5] [6] [7] [8][9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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