1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Mar 30, 2017 #1


    User Avatar

    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

    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


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member
    2017 Award

    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


    User Avatar

    Thanks Orodruin!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted