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Test Question on Inverses

  1. Oct 17, 2015 #1
    How would you answer the following question?

    If ##(f\circ g)(x)=x##, then ##g## is the inverse function of ##f##. True/False?

    This is on a test that I gave and I now think it is a bad question, but I'm tired and I want to hear some other people's impressions. The wording of the question is exactly what I wrote above.
     
  2. jcsd
  3. Oct 17, 2015 #2

    symbolipoint

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    It is a good question, very fair. Independent variable is x. g is a function of x, and f is a function of x. Think what the hypothesis says. Put x into g and then put g into f; and the result is x, the number that you started with. Function f reversed what function g did. INVERSE!

    If you give a function a number x, and the result becomes x, then obviously the function gives you exactly what you gave it. That is what a function composed with its inverse does.
     
  4. Oct 17, 2015 #3

    micromass

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    It's a very good question.
     
  5. Oct 18, 2015 #4
    The reason I have an issue is the example ##f(x)=x^2## and ##g(x)=\sqrt x##. For all ##x## in the domain of ##(f\circ g)(x)## the statement is true because the composition is only defined when ##x\geq0##. However, ##f## is not the inverse of ##g##. I had this on my test for a few years for the reasons stated, but I am not convinced that the implication hold in this direction. That is, I think if ##f## and ##g## are inverses, the equation holds, but if the equation holds, ##f## and ##g## are only inverses given certain qualifications about the domains.
     
  6. Oct 18, 2015 #5

    Mark44

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    That was also the counterexample I thought of, as well. If f and g are inverses, the composition should be commutative, but with these two functions, ##(g \circ f)(x) \neq x##.
     
  7. Oct 18, 2015 #6

    symbolipoint

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    Good catch! We must distinguish between inverse relations and inverse functions.
     
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