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Function Composition

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Functions

  • Recall: what is a function?

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Functions

  • Recall: what is a function?
  • A rule from one set to another set (not necessarily different sets!)
  • Can only have (at most!) one output for any input
  • A function is an action

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Functions

  • Functions can be simple
    • Add one to the input number
  • Functions can be complicated
    • Find the nth prime number
  • Functions can be really complicated
    • Determine the position of all particles in the universe for any time.

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Chain Functions Together

  • Like in life, functions (actions) can be chained together
  • Called composition

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  • Range of output becomes domain of next input

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Scenarios

  • Imagine two activities where the order of the activity matters
    • “I do … then I do …”
  • Imagine two activities where the order of the activity doesn’t matter
    • “I do … then I do …”
  • Share in your groups then as whole class

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Scenario Where Order Matters

  • Let f(x) be “leave house at time f given day x”
      • f(August 5) = 9:30am
      • f(Sept 7) = 6:00am
  • Let g(x) be “arrive at school at time g given time x”
      • g(6:00am) = 6:10am
      • g(7:40am) = 8:00am

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Scenario Where Order Matters

  • Let f(x) be “leave house at time f given day x”
  • Let g(x) be “arrive at school at time g given time x”
  • What makes sense?
    • f(g(x))
    • g(f(x))

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Scenario Where Order Matters

  • Let f(x) be “leave house at time f given day x”
  • Let g(x) be “arrive at school at time g given time x”
  • What makes sense?
    • f(g(x)) - doesn’t make sense
    • g(f(x)) - makes sense
  • Output of f(x) is time which is input of g(x)
  • Composition is from the inside out
  • g(f(x)) is only one that makes sense

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Function Composition

Notation:

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  • Evaluate from inside out
  • g first then f
  • Range of g becomes domain of f

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Examples

Let

Evaluate:

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Examples

  • Write the functions using composition
  • Make new function names for the compositions

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  • How can you check if you’re correct?

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Transformations as Composition

  • Compare
  • Against
  • What is the same? What is different?
  • Compare
  • Against
  • What is the same? What is different?

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