Composite functions

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Introduction
Lessons
  1. What is a "Composite Function"?
  2. Quick review on basic operations with functions.
  3. Composition of Functions: putting one function inside another function!
Examples
Lessons
  1. Introduction to Composite Functions
    If f(x)=4x5f(x)= 4x-5
    g(x)=8x2+6g(x)=8x^2+6
    Determine
    1. (fg)(x)(f\circ g)(x)
    2. (gf)(x)(g\circ f)(x)
    3. (ff)(x)(f\circ f)(x)
    4. (gg)(x) (g\circ g)(x)
    5. (fg)(2) (f\circ g)(2)
      evaluate in two different ways
  2. Evaluate a Composite Function Graphically
    Use the graphs of f(x)f(x) and g(x)g(x) to evaluate the following:


    Composite functions
    1. f(g(4))f(g(-4))
    2. f(g(0))f(g(0))
    3. g(f(2))g(f(-2))
    4. g(f(3))g(f(-3))
  3. Determine the Composition of Three Functions
    Use the functions f(x)=3x,f(x)=3x,
    g(x)=x7g(x)=x-7
    and h(x)=x2h(x)=x^2
    to determine each of the following:
    1. (fgh)(x)(f\circ g\circ h)(x)
    2. g(f(h(x))) g(f(h(x)))
    3. f(h(g(x))) f(h(g(x)))
    4. (hgf)(x)(h\circ g\circ f)(x)
  4. Determine the Original Functions From a Composite Function
    If h(x)=(fg)(x)h(x)=(f\circ g)(x)
    determine f(x)f(x)
    and g(x)g(x)
    1. h(x)=(7x5)34(7x5)+1 h(x)=(7x-5)^3-4(7x-5)+1
    2. h(x)=4x39 h(x)=\sqrt{4x^3-9}, give two possible sets of solutions
  5. Composite Functions with Restrictions
    Consider f(x)=x13f(x)=\sqrt{x-13}
    and g(x)=x2+4g(x)=x^2+4 , for each of the function below:
    1. Determine:
      i)
      (fg)(x)(f\circ g)(x)
      ii)
      (gf)(x)(g\circ f)(x)
    2. State the domain and range of:
      i)
      f(x)f(x)
      ii)
      g(x)g(x)
      iii) (fg)(x)(f\circ g)(x)
      iv)
      (gf)(x)(g\circ f)(x)
    3. Sketch the graph of:
      i)
      f(x)f(x)
      ii)
      g(x)g(x)
      iii) (fg)(x)(f\circ g)(x)
      iv)
      (gf)(x)(g\circ f)(x)
  6. Composite Functions with Restrictions
    Consider f(x)=x13f(x)=\sqrt{x-13}
    and g(x)=x2+4g(x)=x^2+4 , for each of the function below:
    i) state the domain and range
    ii) sketch the graph.
    1. f(x) f(x)
    2. g(x) g(x)
    3. (fg)(x) (f\circ g)(x)
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