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Proving a cubic is surjective.

  1. Mar 15, 2015 #1
    1. The problem statement, all variables and given/known data
    $$f:\mathbb{R}\rightarrow\mathbb{R}~~\text{where}~~f(x)=x^3+2x^2-x+1$$
    Show if f is injective, surjective or bijective.
    2. Relevant equations


    3. The attempt at a solution

    f is obviously not injective (and thus not bijective), one counter example is x=-1 and x=1.

    I can see from the graph of the function that f is surjective since each element of its range is covered.
    But im not sure how i can formally write it down. If i try this;

    ##\text{let}~y\in\mathbb{R}## and then try to find which ##x\in\mathbb{R}## maps to y then the algebra is going to get really messy and it would take a long while. Is there a faster and easier way of showing f is injective? Can i draw the function and show that all of the co-domain is covered, or is this not formal enough?

    Thank you for your time.
     
  2. jcsd
  3. Mar 15, 2015 #2

    RUber

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    Homework Helper

    I think that stating that the function is continuous and tends toward plus or minus infinity for large arguments should be sufficient. You might need to put a little more math and logic into it, but that is the simple argument.
     
  4. Mar 15, 2015 #3

    Dick

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    Science Advisor
    Homework Helper

    Use the Intermediate Value Theorem. http://en.wikipedia.org/wiki/Intermediate_value_theorem
     
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