Composite functions

Composite functions

Lessons

  • 1.
    What is a “Composite Function”?
    a)
    Quick review on basic operations with functions.

    b)
    Composition of Functions: putting one function inside another function!


  • 2.
    Introduction to Composite Functions
    If f(x)=4x5f(x)= 4x-5
    g(x)=8x2+6g(x)=8x^2+6
    Determine
    a)
    (fg)(x)(f\circ g)(x)

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

    c)
    (ff)(x)(f\circ f)(x)

    d)
    (gg)(x) (g\circ g)(x)

    e)
    (fg)(2) (f\circ g)(2)
    evaluate in two different ways


  • 3.
    Evaluate a Composite Function Graphically
    Use the graphs of f(x)f(x) and g(x)g(x) to evaluate the following:


    Composite functions
    a)
    f(g(4))f(g(-4))

    b)
    f(g(0))f(g(0))

    c)
    g(f(2))g(f(-2))

    d)
    g(f(3))g(f(-3))


  • 4.
    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:
    a)
    (fgh)(x)(f\circ g\circ h)(x)

    b)
    g(f(h(x))) g(f(h(x)))

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

    d)
    (hgf)(x)(h\circ g\circ f)(x)


  • 5.
    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)
    a)
    h(x)=(7x5)34(7x5)+1 h(x)=(7x-5)^3-4(7x-5)+1

    b)
    h(x)=4x39 h(x)=\sqrt{4x^3-9}, give two possible sets of solutions


  • 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:
    a)
    Determine:
    i)
    (fg)(x)(f\circ g)(x)
    ii)
    (gf)(x)(g\circ f)(x)

    b)
    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)

    c)
    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)