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Counter example to: if f*g is inv then f & g are inv

  1. Mar 30, 2017 #1

    eq1

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    1. The problem statement, all variables and given/known data

    Provide a counter example to the false assertion:
    Suppose that f and g are functions and f ◦ g is invertible. Then f and g are invertible.

    2. Relevant equations

    Definitions:
    An invertible function is 1-1 and onto
    If the image of g is not contained in the domain of f then f ◦ g is not a legal expression.

    3. The attempt at a solution

    Let f(x)=x^2 over R and g(x)=sqrt(x) over x>0|R, then f(g(x))=x which is 1-1 and onto over x>0|R and is therefore invertible.
    f(x) is not 1-1 over R thus it is not invertible providing the counter example to f and g are invertible.

    My question: The function I defined as f is not invertible but the composition works over a subset of f's domain due to g. Does this mean that f's invertibility should only be considered over the reduced domain too?

    I suspect not because I provided that exact wording of the problem and it says I only need to provide a function definition which is not invertible on its own but which is invertible in a composition.

    I have a strong hunch I am over thinking this...
     
  2. jcsd
  3. Mar 30, 2017 #2

    Orodruin

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    You are doing it right. The way of getting a non-invertible function f is to consider a function that is invertible when restricted to the image of g but not invertible on its own domain. For this to be the case you need a g whose image is not the entire domain of f.
     
  4. Mar 30, 2017 #3

    eq1

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    Thanks Orodruin!
     
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