Mathematics and Logic - Skill and Concept Development

Functions with finite domains

Domains and Ranges of functions y = f(x) may be defined as follows when y and x denote real numbers.

Definition: The domain(f) of the function f is the set of real numbers x for which f(x) is defined.

Definition: The range(f)  of the function f is the set of real numbers y for which there is at least one number x in the domain of f such that y = f(x).

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

arrow diagrams, tables of values, graphs and the vertical line rule (where applicable), graphs and the horizontal line rule (where applicable), ...

with arrow diagrams

A function, dependency,  map or assignment f may defined by arrow diagram.

the arrows say f(1) = a f(2) = c f(3) = a f(4) = b f(5) = c

The domain of f,

 domain( f) ={1,2,3,4,5}= set of values x for which f(x) is defined.

The range of f,

 range (f) ={a,b,c}.

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 tables

We 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. 

If you give two different ways to compute a function, both ways when applicable should give the same result.  Above the arrow diagram and both tables agree for each item in the domain   From a table or from the arrow diagram,  f(3) = a.

The domain of f is still is the set of points {1, 2, 3, 4, 5} and the range of f is still the set of letters {a, b, c}

Another Table Example

Here 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}

h(0) = 1
h(1) = 2.6
h(2) = 4.3
h(3) = 5.8
h(4) = 7.4
h(5) = 9
h(6) = 10.6

The domain of h is set of numbers

{0, 1, 2, 3, 4, 5 ,6}

in the first row.

The range of h is the set of numbers

{1, 2.6, 4.2, 5.8, 7.4, 9, 10.6}

 in the second row.

Yet another table example

A table of values

in which there is no duplicate numbers or objects in the x-row gives a function f with

domain(f) = {1, 2, 3, 4}

The range of f,

range(f) = set of all possible y-values
              = {5, 3, -1, 4}

In this example

f(1) = 5
f(2) = 3
f(3) = -1
f(4) = 4

List Method

A 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 General

A function f defined for a set of distinct values x₁, x₂, ... x_n. by specifying its values y₁, y₂, ... y_n at those numbers, so that 

f(x₁) =  y₁, f(x₂) =  y₂,  ... f(x_n) =  y_n,

Here the domain of definition of f,

Domain (f) = { x₁, x₂, ... x_n.}

 is a finite set. The range of f

Range (f) = { y₁, y₂, ... y_n.}

is a finite set. (Remember to eliminate duplicate values of y so that elements of the range are not listed twice.)

Using ordered pairs

A 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

f(1) = 3.4;  f(2.5) =  4;  f(2.1) = 5 and f(-1) = 8.

The function domain, the set of items for which is defined, is 

domain (f) = { 1, 2.5, 2.1, -1}

Plotting the ordered pairs gives the graph of f. 

The set of points 

f = {  (1,3.4),  (2.5,  4),  (2.1, 5),  (-1, 8) } 

provides the graph of f.  So we may write

f = {  (1,3.4),  (2.5,  4),  (2.1, 5),  (-1, 8) } = graph(f)

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 plane

Here 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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