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High School Mathematics - 2
4.12 Representing Functions as Graphs

Ordered Pairs

An ordered pair is a pair of objects whose components occur in a special order, are separated by a comma and enclosed in a parentheses. In the ordered pair (a,b), a is called the first component and b is called the second component.

Note 1.(a,b) is not the same as {a,b}. The former denotes an ordered pair whereas the latter denotes a set.
Note 2.Ordered pairs may have the same first and second component like (5,5) etc.
Note 3.Two ordered pairs of numbers are defined to be equal when the respective first and second components are equal. Thus (-3, 8) = (-18/6, 24/3) but (2, 3) (3, 2).


Cartesian product

Consider the two sets A and B. The set of all possible ordered pairs (x, y) such that the first component x of ordered pairs is an element of A and the second component y is an element of B, is called the Cartesian product of sets A and B. It is defined as A ~ B which reads "A cross B".
For example if A = {1, 4} and B = {6, 4, 5}, then
A ~ B = {(1,6), (1,4), (1,5), (4, 6), (4, 4), (4, 5)}


Example If g(x) = 3x2-2x+1, find each of the following :
  1. g(0)
    g(0) = 3.(0)2-2.0+1
    g(0) = 1

  2. g(-1)
    g(-1) = 3.(-1)2-2.(-1)+1
    g(-1) =3+2+1
    g(-1) = 6

  3. g(3)
    g(3) = 3.(3)2-2.(3)+1
    g(3) =27-6+1
    g(3) = 22

  4. g(-x)
    g(-x) = 3.(-x)2-2.(-x)+1
    g(-x) = 3x2+2x+1

  5. g(1-t)
    g(1-t) = 3.(1-t)2-2.(1-t)+1
    g(1-t) = 3t2-6t+3+2t-2+1
    g(1-t) = 3t2-4t+2

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