** Fraction Maker. Create Visual Models of fractions and save as images to desktop!


A+    A−    B  
Home
Algebra
Math Games
  • Decimals in Space
  • Fraction Balls
  • Integers in Space
  • Math Man
  • Number Balls
  • Geometry
    Interactive
    Trigonometry
    Jobs
  • Tutoring jobs
  • New York Tutoring Jobs
  • White Plains, NY
  • Westchester County, NY
  • Chicago Math Jobs
  • Philadelphia
  • Teacher Resources
    On FaceBook!

    Composition of Functions

    Functions inside Functions

    A composition of functions occurs when you insert one function into another. In effect, the range of the one function becomes the domain of the second. The notation for composition of functions is either
    • f(g(x))
    • f º g(x)
    The composition of function notation on the left is the composition.
    f(g(x)) = f º g(x)
    However it should be noted that f(g(x)) ≠ g(f(x))
    Pictured on the right is a diagram of the role of the domain and range of each function in the composition of f(g(x)).


    The Commutative Law and Compositions



    Compositions of Functions are a well known example of a process that is not commutative. In other words, f(g(x)) ≠ g(f(x)). The examples below illustrate that although sometimes composition of functions will preserve commuativity (problems 1 and 2), this commutativity is not always true (see problems 3 and 4)
    Practice Problems: Composition of Functions

    The two functions t(x) and v(x) are defined below.
    t(x) = 3x – 1 v(x) = x²+1


    Practice problem 1)
    Evaulate the composition of functions v(t(1))
    Answer


    Practice problem 2)
    Evaulate the composition of functions t(v(1))
    Answer
    t(x) = 3x – 1 v(x) = x²+1



    Are compositions of functions commutative?




    From practice problems 1 and 2, it appears that compositions of functions are commutative, but that is not always true. Compositions of functions are not , as a rule, commutative. In fact, they very rarely happen to be commutative. For example look at the next two compositions and you will see for yourself!


    Practice problem 3)
    Evaulate the composition of functions v(t(2))
    Answer


    Practice problem 4)
    Evaulate the composition of functions t(v(2))
    Answer

    Top