Function and Dependency Definition - More Ways
If the value of a number or quantity y depends on and is unquely determined by the values of numbers a, b, c and so on mathematicians will write y = f(a,b,c, ...) to indicate this dependence where the three dots (ellipsis) stand for the other numbers on which y depends. Another letter except y, a, b and c may be used in place of f in the foregoing function notation y = f(a,b,c, ...) for this dependence.
In the previous lesson, we say how to describe a dependence or function y = f(a,b,c) by a formula with algebraic meaning and sometimes with or without geometric. But we can also describe how to compute the value of the dependent number or quantity y from the values of other numbers and quantities using arrow diagrams, tables of values, graphs and the vertical line rule (where applicable), and graphs and the horizontal line rule (where applicable), and many more ways.
Note: If a dependency is described in more than one way, the different ways must be consistent. That is given the values of the independent values a, b and c, etc, they must give the same value for the dependent variable y.
Definition of Domains and Ranges of functions y = f(x)
- 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).
Definition with arrow diagrams
A function, dependency, map or assignment f may defined by arrow diagram.
 |
the arrows say 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.
Definition with Tables, Example 1
We could have defined the previous function with either one of the two
tables, horizontal
| x | 1 | 2 | 3 | 4 | 5 |
| f(x) | a | c | a | b | c |
or vertical
| x | f(x) |
| 1 | a |
| 2 | c |
| 3 | a |
| 4 | b |
| 5 | c |
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}
Definition with Tables, Example Two
Here we use a table to define h(x).
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | input |
| h(x) | 1 | 2.6 | 4.2 | 5.8 | 7.4 | 9 | 10.6 | output |
The table defines (says how to compute) a function h. Here
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.
Definition with Table of values, Example Three
A table of values
| x | 1 | 2 | 3 | 4 |
| y | 5 | 3 | -1 | 4 |
in which there is no duplicate numbers or objects in the x-row (or column) give 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.
Definition at a list of points
using equations, Example One
f(2) = 3, f(4) =-11 f(8) = 2
The foreging equations imply a function f with domain {3, 4, 8} and range {3, -11, 2}
Definition at a list of points
Example Two
Generalization of Example One
A function f defined for a set of distinct values x1, x2, ... xn. by specifying its values y1, y2, ... yn at those numbers, so that
f(x1) = y1, f(x2) = y2, ... f(xn) = yn,
Here the domain of definition of f,
Domain (f) = { x1, x2, ... xn.}
is a finite set. The range of f
Range (f) = { y1, y2, ... yn.}
is a finite set. (Remember to elimiinate duplicate values of y so that elements of the range are not listed twice.)
Examples using finite sets of 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 that sends 1 to 3.4; 2.5 to 4, 2.1 to 5 and -1 to 8. That is
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. More should be said on that later.
The General Case in the coordinate plane IR2
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.
The set S may be given by a list or by ordered plots in the plane.
A More General Case
Set of Order Pairs in a set product A x B
Suppose a set S given by a list of ordered pairs (x,y) from the set productA x B = {(a,b) | a in A and b in B} = set of all ordered pairs (a,b) where the first element a belongs to A and the second element b belongs to B
A x B = {(a,b) | a in A and b in B}
in which each source element x appears in at most one ordered pair of S (no duplication of first element of ordered pairs). In this case, when the set of points
{(x,y) in A x B | x = a }
intersects the set S at a point (a,b), the associated rule S puts f(a) = b. Here A and B need not be subsets of the real number line.
Vocabulary: A = source set and B = target set.
Problem: What are the domains and ranges of f. Answer that after you have read about the set viewpoint or codification of what is a function.