Concepts of Relations and Functions and How They are Represented

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Concepts of Relations and Functions and How They
are Represented

• Functions are used by mathematicians and
  scientists to describe relationships between
  variable quantities
• Play a central role in calculus and its
  applications
• Use paired data

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Tables and Scatter Plot
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Old Faithful Eruptions Scatter Plot
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Line graph join the successive points
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Histogram/Bar Graph
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Functions

• Tables, graphs, and equations
• Provide three methods for describing how one
  property depends on another
• Tables - numerical Graphs - visual
• Equations - algebraic

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A relation is a function if for each x there is
one and only one y.
A relation is a one-to-one if also for each y
there is one and only one x.
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To be one-to-one, a function must pass the
horizontal line test as well as the vertical line
test.
one-to-one
not one-to-one
not a function
(also not one-to-one)
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If a variable y depends on a variable x in such a
way that each value of x determines exactly one
value of y, then we say that y is a function of x.
A function f is a rule that associates a unique
output with each input. If the input is denoted
by x, then the output is denoted by f(x) (read f
of x).
Functions are represented four basic ways 1)
Numerically by tables 2) Geometrically by
graphs 3) Algebraically by formulas 4) Verbally
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Curve fitting
Converting numerical representations of functions
into algebraic formulas
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Discrete vs Continuous Data

• Discrete Data Data that makes discrete jumps.
  Data represented by scatter plots consisting of
  isolated points. Data that has a finite number of
  values and there is space on a number line
  between 2 possible values. Usually whole
  numbers.
• Continuous Data Data that has values that vary
  continuously over an interval. Data that is
  continuous and unbroken curves. Usually a
  physical measurement, can increase/decrease in
  minutely small values.

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• Classify each set of data as discrete or
  continuous.
• 1) The number of suitcases lost by an airline.
• Discrete. The number of suitcases lost must be a
  whole number.
• 2) The height of corn plants.
• Continuous. The height of corn plants can take on
  infinitely many values (any decimal is possible).
  
• 3) The number of ears of corn produced.
• Discrete. The number of ears of corn must be a
  whole number.

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Classify each set of data as discrete or
continuous.

• 4) The number of green MM's in a bag.
• Discrete. The number of green MM's must be a
  whole number.
• 5) The time it takes for a car battery to die.
• Continuous. The amount of time can take on
  infinitely many values (any decimal is possible).
  
• 6) The production of tomatoes by weight.
• Continuous. The weight of the tomatoes can take
  on infinitely many values (any decimal is
  possible).

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Homework

• Using functions and the analysis of Graphical
  Information
• P22 1 8, 10