Composite functions

Lessons

• Introduction
  What is a "Composite Function"?
  a)

  Quick review on basic operations with functions.
  
  
  b)

  Composition of Functions: putting one function inside another function!
  
  

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• 1.
  Introduction to Composite Functions
  If $f(x)= 4x-5$
  

  $g(x)=8x^2+6$
  Determine
  
  a)

  $(f\circ g)(x)$
  
  
  
  b)

  $(g\circ f)(x)$
  
  
  
  c)

  $(f\circ f)(x)$
  
  
  
  d)

  $(g\circ g)(x)$
  
  
  
  e)

  $(f\circ g)(2)$

  evaluate in two different ways
  
  
  

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• 2.
  Evaluate a Composite Function Graphically
  Use the graphs of $f(x)$ and $g(x)$ to evaluate the following:
  

  
  
  
  
  a)

  $f(g(-4))$
  
  
  
  b)

  $f(g(0))$
  
  
  
  c)

  $g(f(-2))$
  
  
  
  d)

  $g(f(-3))$
  
  
  

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• 3.
  Determine the Composition of Three Functions
  Use the functions $f(x)=3x,$

  $g(x)=x-7$

  and $h(x)=x^2$
  to determine each of the following:
  a)

  $(f\circ g\circ h)(x)$
  
  
  
  b)

  $g(f(h(x)))$
  
  
  
  c)

  $f(h(g(x)))$
  
  
  
  d)

  $(h\circ g\circ f)(x)$
  
  
  

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• 4.
  Determine the Original Functions From a Composite Function
  If $h(x)=(f\circ g)(x)$

  determine $f(x)$

  and $g(x)$
  
  a)

  $h(x)=(7x-5)^3-4(7x-5)+1$
  
  
  
  b)

  $h(x)=\sqrt{4x^3-9}$, give two possible sets of solutions
  
  
  

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• 5.
  Composite Functions with Restrictions
  Consider $f(x)=\sqrt{x-13}$

  and $g(x)=x^2+4$, for each of the function below:
  a)

  Determine:
  

  i)

  $(f\circ g)(x)$
  

  ii)

  $(g\circ f)(x)$
  
  
  
  b)

  State the domain and range of:
  

  i)

  $f(x)$
  

  ii)

  $g(x)$
  

  iii) $(f\circ g)(x)$
  

  iv)

  $(g\circ f)(x)$
  
  
  
  c)

  Sketch the graph of:
  

  i)

  $f(x)$
  

  ii)

  $g(x)$
  

  iii) $(f\circ g)(x)$
  

  iv)

  $(g\circ f)(x)$
  
  
  

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• 6.
  Composite Functions with Restrictions
  Consider $f(x)=\sqrt{x-13}$

  and $g(x)=x^2+4$, for each of the function below:
  

  i) state the domain and range
  

  ii) sketch the graph.
  a)

  $f(x)$
  
  
  b)

  $g(x)$
  
  
  c)

  $(f\circ g)(x)$
  
  

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