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Math Help - Domain and range

  1. #1
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    Domain and range

    obtain the domain and range of ln(g(x))??


    g(x)=6+4x-2x2




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  2. #2
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    Hi

    g(x)=-2x+4x+6 = -2(x+1)(x-3)
    It must be strictly positive because ln(a) is defined for a>0
    The domain is therefore ]-1,3[

    g(x)=-2x+4x+6 is a parabola oriented towards negative y
    Maximum is obtained for x=1 and g(1)=8
    Range is therefore ]-oo,3ln(2)]
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  3. #3
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    Quote Originally Posted by running-gag View Post
    Hi

    g(x)=-2x+4x+6 = -2(x+1)(x-3)
    It must be strictly positive because ln(a) is defined for a>0
    The domain is therefore ]-1,3[

    g(x)=-2x+4x+6 is a parabola oriented towards negative y
    Maximum is obtained for x=1 and g(1)=8
    Range is therefore ]-oo,3ln(2)]
    Can you explain in details How can you find the range? ,I don't understand it
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  4. #4
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    OK
    ln(g(x)) is defined on ]-1,3[

    On ]-1,3[ g(x) is increasing between -1 and 1 and decreasing between 1 and 3
    Therefore ln(g(x)) is increasing between -1 and 1 and decreasing between 1 and 3

    g(-1)=0 => ln(g(x)) has -oo for limit in -1
    g(1)=8 => ln(g(1)) = ln(8) = ln(2^3) = 3 ln(2)
    g(3)=0 => ln(g(x)) has -oo for limit in 3
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  5. #5
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    Quote Originally Posted by running-gag View Post
    OK
    ln(g(x)) is defined on ]-1,3[

    On ]-1,3[ g(x) is increasing between -1 and 1 and decreasing between 1 and 3
    Therefore ln(g(x)) is increasing between -1 and 1 and decreasing between 1 and 3

    g(-1)=0 => ln(g(x)) has -oo for limit in -1
    g(1)=8 => ln(g(1)) = ln(8) = ln(2^3) = 3 ln(2)
    g(3)=0 => ln(g(x)) has -oo for limit in 3

    Thank you very much for Explaining
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  6. #6
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    when I study again I have question in my mind

    why we don't say that the range (0,ln8]
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  7. #7
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    Because when x is close to -1 g(x) is close to 0 and we know that ln has a limit -oo in 0. Therefore ln(g(x)) has -oo as limit when x is close to -1

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  8. #8
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    thank you very much .. now range will be very clear for me
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