doing operations on both sides

Discussion in 'Undergraduate Math' started by G Patel, Aug 20, 2007.

  1. G Patel

    G Patel Guest

    In the past I've done things like taking the derivative/limit of both
    sides of an equation without thinking much about why I could do this

    Recently I got to thinking about "doing an operation on both sides."
    Which, in general, preserve the equality after the operation? I know
    there is an issue with operations like "squaring both sides" where the
    new equation might have extra roots.

    What type of operations can cause problems? Is there a general rule
    on this?
    G Patel, Aug 20, 2007
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  2. Recently I got to thinking about
    In general, you want reversability.

    Squaring both sides can lead to problems because
    although a=b => a^2=b^2, the reverse implication
    is invalid, because squaring is not reversable.
    It's not one-to-one.
    riderofgiraffes, Aug 20, 2007
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  3. G Patel

    -kk- Guest

    a=b => f(a)=f(b)(obviously).
    We can only assume the reverse implication [ f(a)=f(b) => a=b ]
    when f is injective.

    For the example you mention, f(x)=x^2: this function is not injective
    when x is any real number. However, we can make any function injective
    by restricting the domain (and any function surjective by restricting
    the codomain). For the above example we can (for example) restrict
    the domain to the nonnegative reals (or integers etc). The 'rule'
    here is injectivity.

    -kk-, Aug 22, 2007
  4. G Patel

    Dear Leader Guest

    you mean function? single value, one y for one x
    i.e. not have multi-value, like two y for one x
    Dear Leader, Aug 22, 2007
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