Functions 2.1 (A)

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Functions 2.1 (A)
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What is a function?

• Rene Descartes (1637) Any positive integral
  power of a variable x.
• Gottfried Leibniz (1646-1716) Any quantity
  associated with a curve
• Leonhard Euler (1707-1783) Any equation with 2
  variables and a constant
• Lejeune Dirichlet (1805-1859) Rule or
  correspondence between 2 sets

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What is a relation?

• Step Brothers?
• Math Definition
• Relation A correspondence between
• 2 sets
• If x and y are two elements in these sets, and if
  a relation exists between them, then x
  corresponds to y, or y depends on x
• x ? y or (x, y)

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Example of relation

• Names Grade on Ch. 1 Test
• Buddy A
  
• Jimmy B
• Katie C
• Rob

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Dodgeball Example

• Say you drop a water balloon off the top of a 64
  ft.
• building. The distance (s) of the dodgeball
  from the ground after t seconds is given by the
  formula
• Thus we say that the distance s is a function of
  the time t because
• There is a correspondence between the set of
  times and the set of distances
• There is exactly one distance s obtained for any
  time t in the interval

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Def. of a Function

• Let X and Y be two nonempty sets. A function
  from X into Y is a relation that associates with
  each element of X exactly one element of Y.
• Domain A pool of numbers there are to choose
  from to effectively input into your function
  (this is your x-axis).
• ?
• The corresponding y in your function is your
  value (or image) of the function at x.
• ?
• Range The set of all images of the elements in
  the domain (This is your y-axis)

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Domain/Range Example

• Determine whether each relation represents a
  function. If it is a function, state the domain
  and range.
• a) (1, 4), (2, 5), (3, 6), (4, 7)
• b) 1, 4), (2, 4), (3, 5), (6, 10)
• c) -3, 9), (-2, 4), (0, 0), (1, 1), (-3, 8)

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Practice

• Pg. 96 2-12 Even

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Function notation

• Given the equation
• Replace y with f(x)
• f(x) means the value of f at the number x
• x independent variable
• y dependent variable

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Finding values of a function

• For the function f defined by
• evaluate
• a) f(3)
• b) f(x) f(3)
• c) f(-x)
• d) f(x)
• e) f(x 3)
• f)

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Practice 2

• Pg. 96 14, 18, 20

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Implicit form of a function

• Implicit Form

• Explicit Form

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Determine whether an equation is a function

• Is a function?

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Finding the domain of a function

• Find the domain of each of the following
  functions

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Tricks to Domain

• Rule 1
• If variable is in the denominator of function,
  then set entire denominator equal to zero and
  exclude your answer(s) from real numbers.
• Rule 2
• If variable is inside a radical, then set the
  expression greater than or equal to zero and you
  have your domain!

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Practice 3

• Pg. 96 22-46 E