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MAT091 :: Lecture Note :: Week 07
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Overview
Assignment(s)

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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Increasing and Decreasing Functions

Functions are sometimes categorized as either increasing or decreasing.

An increasing function is a function whose output values increase (i.e. get larger) when the inputs increase.

   f(0) = 5
   f(1) = 10
   f(2) = 15
   f(3) = 20

   f(x) appears to be an increasing function

A decreasing function is a function whose output values decrease (i.e. get smaller) when the inputs increase.

   g(0) = 100
   g(1) = 90
   g(2) = 80
   g(3) = 70

   g(x) appears to be a decreasing function

A constant function f(x) = k (where 'k' is a constant) is neither increasing nor decreasing.

The identity function f(x) = x is an increasing function.

Some functions can be both increasing and decreasing. For example, over an interval A, a function might be increasing, while it decreases over an interval B. Review the absolute value function.

   abs(x) is decreasing from (-INF, 0) 
   abs(x) is zero at 0
   abs(x) is increasing from (0, INF)

Technical definitions for increasing and decreasing functions.

   increasing function:
   ====================
   Function f(x) increases on an interval I if f(b) > f(a) 
                 for all b > a, where a,b are in I.

   decreasing function:
   ====================
   Function f(x) decreases on an interval I if f(b) < f(a) 
                 for all b > a, where a,b are in I.

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Linear Functions

M-W.com defines linear as follows.

   "of, relating to, resembling, or having a graph that 
    is a line and especially a straight line"

   [...and...]

   "having or being a response or output that is directly 
    proportional to the input"

Observe that the word linear contains the word line.

A linear function is a function is a "first degree polynomial function of one variable. These functions are called 'linear' because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line."

The following are equations for a line.

   slope-intercept form:  y = mx + b
   ... used when slope and y-intercept are known

   point-slope form:  y - y1 = m(x - x1)
   ... where (x1, y1) is a point on the line having slope m

   standard form:  Ax + By = C

Slope is another way of saying "rate of change" and linear functions have a constant rate of change.

Slope is often described as the ratio "rise over run."

Linear functions with a positive constant rate of change are increasing functions. Linear functions with a negative constant rate of change are decreasing functions.

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Slope (Linear Functions)

The slope of a line is a measure of its average rate of change (steepness?). The slope also indicates if the line is increasing (uphill?) or decreasing (downhill?). In linear equations, slope is represented by the letter m.

The following is how to find the slope between two unique points on a line.

   point 1:  (x1, y1)
   point 2:  (x2, y2) 

   slope = m = (y2 - y1) / (x2 - x1)

   note:  x1 ≠ x2

Slope represents the ratio of change-in-output over change-in-input.

   m = Δy / Δx    (change in output / change in input)

   Δ is the "delta" character
   Δy is read "change of" (Δy is read "change of" y)
   Δy = y2 - y1

Slope is often described as the ratio "rise over run."


        rise
   m = ------
        run

Some slope notes.

   horizonal lines have slope 0   (y = f(x) = k, where k is a consant)
   vertical lines have undefined slope   (x = k, where k is a consant)
   parallel lines have equal slopes     (m1 = m2)
   perpendicular lines have negative inverse slopes  (m2 = -1/m1)

The following is a verbose form the slope-intercept equation.

   output = slope * input + initial_value_if_any

   ...or...

   output = average_rate_of_change * input + initial_value_if_any

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Intercepts (Linear Functions)

We frequently need to know the points where lines (graphs in general) intersect (cross) the horizonal-axis and vertical-axis (often called the x-intercept and y-intercept, respectively).

The vertical-intercept for a function is found by evaluating the function when its input is zero. The orderd-pair for a vertical-intercept will be (0, something).

The horizontal-intercept for a function is found by setting its output to zero and finding what input value results in that output. The orderd-pair for a horizonal-intercept will be (something, 0).

   f(x) = 5(x) + 3

   vertical-intercept (y-intercept)... 
      set the input to 0 and evaluate:  f(0) = 5(0) + 3 = 3
      vertical-intercept is (0, 3)

   horizontal-intercept (x-intercept)... 
      set the output to 0 and solve:  0 = 5(x) + 3 
                                     -3 = 5(x)
                                   -3/5 = x
      horizontal-intercept is (-3/5, 0)

Generalizations.

   given  f(x) = m(x) + b

     vertical-intercept:  (0, b)        [notice input is 0]
   horizontal-intercept:  (-b/m, 0)     [notice output is 0]

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