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Math Help - Math Help Forum: Are these functions injective/surjective etc?

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    Math Help Forum: Are these functions injective/surjective etc?

    1. Let f : R -> R be the function given by f(x) = x^3 - x^2 - 2x.
    (a) The function f is injective.
    (b) The function f is surjective.
    (c) f^-1({0}) = {0}
    (d) -1 ∈ f(R)

    2. Let f : R2 -> R2 be a function given by f(x1; x2) = (3; x1 - 2*x2)
    (a) The function f is injective
    (b) The function f is surjective
    (c) The image of f is the line de ned by y1 - 2*y2 = 3
    (d) f^-1({(y1; y2) ∈ R2 , y2 = 0}) is the line de ned by x1 - 2*x2 = 0.

    Please help! I don't get it at all....
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    Re: Math Help Forum: Are these functions injective/surjective etc?

    Quote Originally Posted by misqa88 View Post
    1. Let f : R -> R be the function given by f(x) = x^3 - x^2 - 2x.
    (a) The function f is injective.
    (b) The function f is surjective.
    (c) f^-1({0}) = {0}
    (d) -1 ∈ f(R)
    Did you graph this one?
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    Re: Math Help Forum: Are these functions injective/surjective etc?

    yes I did and it looks like a polynomial graph.
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    Re: Math Help Forum: Are these functions injective/surjective etc?

    Quote Originally Posted by misqa88 View Post
    yes I did and it looks like a polynomial graph.
    Look at the domain [-2,2] is it one-to-one?
    For the whole graph, does it pass the horizontal line test?
    Where is the graph zero?
    Does the line y=-1 cut the graph?
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    Re: Math Help Forum: Are these functions injective/surjective etc?

    1) yes it is one-to-one (so the function is injective)
    2) no the graph doesn't pass the horizontal line test
    3) do you mean the pint (0,0)? it's above the graph
    4) yes the line cuts the graph
    Last edited by misqa88; October 25th 2012 at 01:24 PM.
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    Re: Math Help Forum: Are these functions injective/surjective etc?

    Quote Originally Posted by misqa88 View Post
    1) yes it is one-to-one (so the function is injective)WRONG
    2) yes the graph passes the horizontal line test So it is onto.
    3) do you mean the pint (0,0)? it's above the graph NO!
    4) yes the line cuts the graph So -1\in f(\mathbb{R})
    f^{-1}(\{0\})=\{-1,0,2\}
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    Re: Math Help Forum: Are these functions injective/surjective etc?

    Oh sorry! the function is not injective because the y-values are not unique for x-values, but it is subjective because it passes the horizontal line test. Is it always the case? Can I always check the surjectivity of a function like that? Sorry for all those questions but you are the most helpful so far.
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    Re: Math Help Forum: Are these functions injective/surjective etc?

    Quote Originally Posted by misqa88 View Post
    Oh sorry! the function is not injective because the y-values are not unique for x-values, but it is subjective because it passes the horizontal line test. Is it always the case? Can I always check the surjectivity of a function like that? Sorry for all those questions but you are the most helpful so far.
    If every horizontal line intersects the graph at least once the the function is surjective( onto \mathbb{R} )
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    Re: Math Help Forum: Are these functions injective/surjective etc?

    I see, thank you, but what happens with the example 2 when the function looks like that f(x1; x2) = (3; x1 - 2*x2)?
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    Re: Math Help Forum: Are these functions injective/surjective etc?

    1. Let f : R -> R be the function given by f(x) = x^3 - x^2 - 2x.
    (a) The function f is injective.
    (b) The function f is surjective.
    (c) f^-1({0}) = {0}
    (d) -1 ∈ f(R)

    Guys, for surjective or not, find maxima and minima, then see if those are bounded or not (not +infinity and -infinity). If those are bounded, then it is not surjective, otherwise it is surjective.

    Salahuddin
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    Re: Math Help Forum: Are these functions injective/surjective etc?

    Quote Originally Posted by Salahuddin559 View Post
    Guys, for surjective or not, find maxima and minima, then see if those are bounded or not (not +infinity and -infinity). If those are bounded, then it is not surjective, otherwise it is surjective.
    That is not true if the function is not everywhere continuous.
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    Re: Math Help Forum: Are these functions injective/surjective etc?

    That is correct. But in our examples, those are continuous and differentiable, etc... right. No harm of using it as long as those are polynomials, and such.

    Salahuddin
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