Decomposing A Composite Function

Discussion in 'Other Pre-University Math' started by nycmathguy, Aug 3, 2021.

  1. nycmathguy

    nycmathguy

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    Section 1.8
    Question 50

    See attachment.

    I know that (f ° g)(x) = f(g(x)).

    We have f(g(x)) = h(x).

    This leads to f(g(x)) = (1 - x)^3.

    Stuck here....

    20210803_035330.jpg
     
    nycmathguy, Aug 3, 2021
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  2. nycmathguy

    MathLover1

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    This leads to f(g(x)) = (1 - x)^3 => g(x)=1 - x and f(x)=x^3
     
    MathLover1, Aug 3, 2021
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  3. nycmathguy

    nycmathguy

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    How did you conclude that f(x) = x^3?
     
    nycmathguy, Aug 4, 2021
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  4. nycmathguy

    MathLover1

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    f(g(x)) -> f after g(x)
    f(g(x)) = (1 - x)^3=> on the left side, g(x) is inside parentheses and on the right side 1 - x is inside parentheses , means
    g(x)=1 - x which means (1 - x) represents x in f(x)
    then f(x )= x^3
     
    MathLover1, Aug 4, 2021
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  5. nycmathguy

    nycmathguy

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    This is not clear to me. Sorry. I will try two more on my own. If I get it right, I may try a few more samples. If not, I am moving on.
     
    nycmathguy, Aug 4, 2021
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  6. nycmathguy

    MathLover1

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    "Function Composition" is applying one function to the results of another:
    -----> f( ) --------> g( ) ------>
    The result of f() is sent through g()
    It is written: (g º f)(x)
    Which means: g(f(x))

    Example: f(x) = 2x+3 and g(x) = x^2

    "x" is just a placeholder. To avoid confusion let's just call it "input":

    f(input) = 2(input)+3

    g(input) = (input)^2

    (g º f)(x) = g(f(x))

    First we apply f, then apply g to that result:


    -----> (2*input+3 ) --------> (input )^2 ------>

    (g º f)(x) = (2x+3)^2

    your case is in reverse order

    f(g(x)) = (1 - x)^3 =>f(input) = (input)^3 =>"x" is= "input" =>f(x) = (x)^3
     
    MathLover1, Aug 5, 2021
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  7. nycmathguy

    nycmathguy

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    Much better now. I will use new reply to work on two more problems. Work will be posted when time allows.
     
    nycmathguy, Aug 5, 2021
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