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MAT091 :: Lecture Note :: Week 09
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### Ordered Pairs

The inputs and outputs of a function are often represented as ordered pairs (or points).

(input, output) ...or... (x, y) ...or... (x, f(x))
(t, n) ...or... (t, f(t))
(n, o) ...or... (n, f(n))
(a, b) ...or... (a, f(a))

The input is always the first value recorded followed by its respective output.

Data contained in tables can sometimes be represented as ordered-pairs.

input:  month ......... output:  #days

month |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 | 10 | 11 | 12
------------------------------------------------------------------
#days | 31 | 28 | 31 | 30 | 31 | 30 | 31 | 31 | 30 | 31 | 30 | 31

(1, 31)   (2, 28)  (3, 31)  (4, 30)   (5, 31)  (6, 30)
(7, 31)   (8, 31)  (9, 30)  (10, 31)  (11,30)  (12, 31)

#days depends on the month (i.e. the output depends on the input)

The function  f(m)  takes a month as input and outputs
the number of days in that month.

f(m) = n

f(1) = 31
f(2) = 28
f(7) = 31
f(12) = 31

Ordered-pairs are points that can be graphed. The input is along the horizontal-axis and output is along the vertical-axis.

The following ordered-pairs could represent a function because all of the inputs are unique (i.e. different).

(-2, 3)  (0, 4)  (5, -2)  (7, 4)  (11, 7)

The following ordered-pairs could represent a function because all of the inputs are unique although some of the inputs repeat.

(-2, 3)  (0, 4)  (5, -2)  (7, 4)  (11, 7)  (0, 4)

The following ordered-pairs cannot be a function because input 3 produces different outputs (2 and 5).

(-2, 3)  (0, 4)  (-2, 3)  (3, 2)  (0, 4)  (3, 5)

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### Using Tables To Represent Functions

When it comes to graphing functions, tables and other forms of data, input values are scaled along the horizontal-axis and output values are scaled along the vertical-axis.

When it comes to naming variables, the letter x is often used to represent the input variable and the letter y is used to represent the output variable; however, any letters can be used when it comes to naming input and output variables.

Table A
===============================================================
input: year (y)     | 2001 | 2002 | 2003 | 2004 | 2005 | 2007
---------------------------------------------------------------
output: # x-rays (x) |  55     43     81     24     43     61

The # of x-rays is a function of the year; i.e. year is the input
and # of x-rays is the output.  For this function, the input
variable is named 'y' and the output variable is named 'x'.

Table A expressed using function notation.

f(y) = x
f(2001) = 55
f(2002) = 43
f(2003) = 81
...
f(2000) = undefined  [cannot extrapolate]
f(2006) = undefined  [cannot interpolate]

Table B
==============================================================
input: hours (h)     | 1  |  2  |  3  |  4  |  5  |  6  |  7
--------------------------------------------------------------
output: minutes (m)   | 60 | 120 | 180 | 240 | 300 | 360 | 420

Number of minutes is a function of number of hours.
One hour is 60 minutes, two hours is 120 minutes, etc.

Table B expressed using function notation.

g(h) = m = 60(h)
g(1) = 60
g(4) = 240
g(7) = 420
g(2.5) = 150   [interpolation]
g(5.25) = 315  [interpolation]
g(0) = 0       [extrapolation]
g(8) = 480     [extrapolation]

[definition] A function is a "relation for which each element of the domain corresponds to exactly one element of the range."

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