Section 2'1: Properties of Functions

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Section 2.1 Properties of Functions

• Homework 1-65 (pages 59-62)

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Questions?

• Let us use a few minutes to answer any urgent
  questions you may have about last nights
  homework

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Function Notation

• A function is a rule f that assigns to every
  input x one and only one output y.
• For an input x, the output is usually denoted y
  f(x), where f is the name given to the function.
• Do not read the equation above as f times x
  but, rather, as f of x.

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Examples (using people instead of numbers)

• If one refers to xs brother as b(x), that would
  not be a function.

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Examples (using people instead of numbers)

• If one refers to xs brother as b(x), that would
  not be a function.
• When you talk about Davids brother, are you
  talking about Daniel or Michael?

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Examples (using people instead of numbers)

• If one refers to xs brother as b(x), that would
  not be a function.
• When you talk about Davids brother, are you
  talking about Daniel or Michael?
• Also, what if someone does not have a brother at
  all?

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Examples (using people instead of numbers)

• If one refers to xs father as f(x), that would
  be a function.

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Examples (using people instead of numbers)

• If one refers to xs father as f(x), that would
  be a function.
• Everybody has one and only one father.

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Examples with numbers

• f(x) 3

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Examples with numbers

• f(x) 3 is a function.

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Examples with numbers

• f(x) 3 is a function.
• g(x) 3x

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Examples with numbers

• f(x) 3 is a function.
• g(x) 3x is a function.

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Examples with numbers

• f(x) 3 is a function.
• g(x) 3x is a function.
• y f(x) 2x 4

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Examples with numbers

• f(x) 3 is a function.
• g(x) 3x is a function.
• y f(x) 2x 4 is a function.
• We call them

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Examples with numbers

• f(x) 3 is a function (a constant function)
• g(x) 3x is a function
• y f(x) 2x 4 is a function

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Examples with numbers

• f(x) 3 is a function (a constant function)
• g(x) 3x is a function (a linear function)
• y f(x) 2x 4 is a function

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Examples with numbers

• f(x) 3 is a function (a constant function)
• g(x) 3x is a function (a linear function)
• y f(x) 2x 4 is a function (a linear
  function)

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Linear Functions

• f(x) 3 is a function (a constant function)
• g(x) 3x is a function (a linear function)
• y f(x) 2x 4 is a function (a linear
  function)
• The terminology makes sense since the
  relationship between x and y is given by a linear
  equation y mx b.

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Domain and Range

• The set of possible inputs is the domain.
• The set of possible outputs is the range.

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Revisiting the brother function

• If one refers to xs brother as b(x), that would
  not be a function.
• What if someone does not have a brother at all?
• We could define the function b(x) so that the
  domain consists of only people who have brothers.

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Revisiting the brother function

• What if someone has more than one brother?
• Well, define, for example, b(x) to be xs oldest
  brother

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Going back to numerical examples

• Let f(x) x2.
• This is a function. Its domain is all real
  numbers and its range is 0, infinity)
• f(2)
• f(-1)
• f(10)

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Going back to numerical examples

• Let f(x) x2.
• This is a function. Its domain is all real
  numbers and its range is 0, infinity)
• f(2) 4
• f(-1)
• f(10)

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Going back to numerical examples

• Let f(x) x2.
• This is a function. Its domain is all real
  numbers and its range is 0, infinity)
• f(2) 4
• f(-1) 1
• f(10)

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Going back to numerical examples

• Let f(x) x2.
• This is a function. Its domain is all real
  numbers and its range is 0, infinity)
• f(2) 4
• f(-1) 1
• f(10) 100

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A numerical rule defined with words

• g(x) the number which, when squared, yields x
  back.

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A numerical rule defined with words

• g(x) the number which, when squared, yields x
  back.
• Is this a function?

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A numerical rule defined with words

• g(x) the number which, when squared, yields x
  back.
• Is this a function?
• Does every input have an output? (In other words,
  what should be the domain of g?)

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A numerical rule defined with words

• g(x) the number which, when squared, yields x
  back.
• Is this a function?
• Does every input have an output? (In other words,
  what should be the domain of g?)
• g(x) could be a function with domain
  0,infinity). The output of -4,for example,
  could not be defined.

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A numerical rule defined with words

• For any non-negative number x, let g(x) the
  number which, when squared, yields x back.
• Is this a function?

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A numerical rule defined with words

• For any non-negative number x, let g(x) the
  number which, when squared, yields x back.
• Is this a function?
• Does every input have only one input?

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A numerical rule defined with words

• For any non-negative number x, let g(x) the
  number which, when squared, yields x back.
• Is this a function?
• Does every input have only one input?
• As defined, g(4) could be either 2 or -2.

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A numerical function defined with words

• For any non-negative number x, let g(x) the
  non-negative number y which, when squared, yields
  x back. (i.e. y2 x).

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A numerical function defined with words

• For any non-negative number x, let g(x) the
  non-negative number y which, when squared, yields
  x back. (i.e. y2 x).
• Now we recognize this function as g(x) the
  square root of x.

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In this class

• I will pay much more attention to the Domain of
  Functions than to their Ranges.
• Calculating ranges is a little more complicated
  an, in general, does not seem to be worth the
  extra work.

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Also

• Calculating domains will usually boil down to
  answering the question What values of x can we
  not plug in?

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Calculating Domains

• Calculating domains will usually boil down to
  answering the question What values of x can we
  not plug in?
• There will only be two problems we face in this
  class when it comes to domains of functions
  zeroes in the denominator and negative numbers
  under (even) radicals.

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Examples

• The domain of f(x) 1/(x-5) consists of all
  numbers but x 5.
• The domain of g(x) / 2x 1 (the square root of
  2x 1) consists of all numbers bigger than equal
  to -1/2.

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One last thought

• Vertical Line Test.