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Math Help - Testing if an inverse exists

  1. #1
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    Testing if an inverse exists

    The question:
    Show that the function f: R -> R, given by f(x) = x^2, is not one-to-one.

    My attempt:
    According to my text, "A function f is said to be one-to-one if f(a) = f(b) implies that a = b whenever a, b is an element Dom(f).

    So:

    a^2 = b^2
    \sqrt{a^2} = \sqrt{b^2}
    |a| = |b|

    I know by intuition that x^2 is not one to one, however it appears that a does equal b. Or is the fact that each side is the absolute value that the proof fails? Thank you.
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  2. #2
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    Quote Originally Posted by Glitch View Post
    The question:
    Show that the function f: R -> R, given by f(x) = x^2, is not one-to-one.

    My attempt:
    According to my text, "A function f is said to be one-to-one if f(a) = f(b) implies that a = b whenever a, b is an element Dom(f).

    So:

    a^2 = b^2
    \sqrt{a^2} = \sqrt{b^2}
    |a| = |b|

    I know by intuition that x^2 is not one to one, however it appears that a does equal b. Or is the fact that each side is the absolute value that the proof fails? Thank you.
    The fact that, for example, f(-1) = f(1) = 1 clearly demonstrates that f is not 1-to-1.
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  3. #3
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    I know that it isn't one-to-one, I was just trying to apply the formal procedure outlined in my text.

    Now that I think about it, the absolute value does break it since it makes different signed values for a and b equal even if they're not (such as your example). Thanks.
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  4. #4
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    Yes, the crucial point in this problem being that "|a|= |b|" is NOT the same as "a= b"!
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