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Math Help - Left Inverse

  1. #1
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    Left Inverse

    For each of the mappings f given in Excercise 1, determine whether f has a left inverse. Exhibit a left inverse whenever one exists

    b. f(x) = 3x

    I know the function is one to one and has a left inverse as follows

    \begin{array}{rl}<br />
f(m) = f(n) \implies & 3m = 3n\\<br />
\implies & m = n \end{array}

    Based on the lemma we have (lemma 1.24 in our book we know the following)

    If there is an element y \in Z such that f(y)=x, then g(x)= y.
    f(x) = 3x
    f(y) = 3y
    x = 3y
    \frac{x}{3} = y
    g(x) = \frac{x}{3}

    Now I know that is the inverse of the function but is that the inverse only when x is a multiple of 3? and it's an arbitrary a_{0} for all other conditions? How would I write that ...

    A left inverse g: Z \rightarrow Z is \[ g(x) = \left\{ \begin{array}{ll}<br />
\frac{x}{3} & \mbox{if x is a multiple of 3}\\<br />
1 & \mbox{if x is not a multiple of 3}\end{array} \right. \]

    As you can see the problem is essentially solved it's more or less how do I format my answer that would be expected by a professor when I take this course next semester.
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  2. #2
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    Quote Originally Posted by imind View Post
    As you can see the problem is essentially solved it's more or less how do I format my answer that would be expected by a professor when I take this course next semester.
    I doubt that any of us could possibly know what any particular professor expects of you. You did not even tell the name of the course or the level.
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  3. #3
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    It's modern algebra ... 3000 level or junior level ...

    Does anyone know if the above I wrote is actually correct? (the course shouldn't be relevant to if the information is right or not ...)
    Last edited by imind; February 13th 2011 at 07:49 PM. Reason: clarification
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  4. #4
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    Found the answer I needed ... it is indeed

    \[ g(x) = \left\{ \begin{array}{ll}<br />
\frac{x}{3} & \mbox{if x is a multiple of 3}\\<br />
1 & \mbox{if x is not a multiple of 3}\end{array} \right. \]

    1 can be any arbitrary value you choose.
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