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Math Help - onto function

  1. #1
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    onto function

    A function f is from R-------R such that f(x)=(a x^2+6x-8)/(a+6x-8x^2)
    find the value of a for which for which f is onto.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by ayushdadhwal View Post
    A function f is from R-------R such that f(x)=(a x^2+6x-8)/(a+6x-8x^2) find the value of a for which for which f is onto.


    f(x)=y\Leftrightarrow \ldots \Leftrightarrow (a+8y)x^2+(6-6y)x-8-ay=0

    The above equation on x has real solution iff:

    p_a(y)=\Delta=B^2-4AC=\ldots =(9+8a)y^2+(46+a^2)y+9+8a\geq 0

    We have a family of parabolas (a line if a=-9/8). We need to find a such that p_a(y)\geq 0 for all y\in\mathbb{R} . Necessarily a>-9/8 (why?) .

    If a>-9/8 then, f is onto iff p(y_0)\geq 0 being y_0 the "abscissa" of the p_a vertex.

    Let us see what do you obtain.
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  3. #3
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    good evening dear Sir
    sir the point you mention that a>-9/8 means a upward parabola .definitely it has a bend some where .so that will give some minimum value of y.sir i am little bit confused here.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by ayushdadhwal View Post
    good evening dear Sir
    sir the point you mention that a>-9/8 means a upward parabola .definitely it has a bend some where .so that will give some minimum value of y.sir i am little bit confused here.

    And if that minimum is \geq 0 then, p_a(y)\geq 0 for all y\in\mathbb{R} .
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