1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proving functions are surjective

  1. Nov 8, 2013 #1
    prove

    whether or not the following functions are surjective or injective:
    1) [tex] g: \mathbb{R} \rightarrow \mathbb{R}[/tex] [tex] g(x) = 3x^3 - 2x [/tex]

    2)[tex] g: \mathbb{Z} \rightarrow \mathbb{Z}[/tex] [tex] g(x) = 3x^3 - 2x [/tex]


    my working for 1):

    injective: suppose [tex] g(x') = g(x) [/tex] : [tex] 3x'^3 - 2x' = 3x^3 - 2x [/tex] this does not imply [tex] x = x' [/tex] hence not injective
    surjective: need to show [tex] \forall y \in \mathbb{R} \exists x \in \mathbb{R} s.t. g(x) = y [/tex], [tex] y = 3x^3 - 2x [/tex] as cubics always have one real root then as ## y \in \mathbb{R} ## ## \exists x \in g(x) \in \mathbb{R} ## s.t. ## g(x) = y ## therefore it's surjective

    2):
    injective: same as 1:
    surjective: I'm not sure how to phrase this for the integers,

    overall I'm not happy with my proofs, for the injectivity I haven't really shown that x is not equal to x', how would I do it? And for surjectivity I have mainly written it in words, how would I write it out formally for both questions?
     
  2. jcsd
  3. Nov 8, 2013 #2

    FeDeX_LaTeX

    User Avatar
    Gold Member

    Why doesn't it imply that it's not injective? You seem to have skipped a step here.
     
  4. Nov 8, 2013 #3
    I don't really know I just saw it as that :\
     
  5. Nov 8, 2013 #4

    FeDeX_LaTeX

    User Avatar
    Gold Member

    Can you find an example which demonstrates that g is not injective?
     
  6. Nov 8, 2013 #5
    0 and rt(2/3)
     
  7. Nov 10, 2013 #6

    FeDeX_LaTeX

    User Avatar
    Gold Member

    Yes, that works. The function's surjectivity can be verified via the intermediate value theorem.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Proving functions are surjective
  1. Surjective function (Replies: 3)

  2. Surjective functions (Replies: 1)

Loading...