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Math Help - Domain and range

  1. #1
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    Domain and range

    Could you all please help me get started on finding the domain and range of these 2 functions

     g(x) = \sqrt{x^2-2x-5}

    and

     g(x) = (25+4x-x^2)^\frac{1}{2}

    Thank you

    Update: i can find the domains, but the ranges are giving me trouble
    Last edited by silencecloak; May 12th 2008 at 05:51 PM.
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    Quote Originally Posted by silencecloak View Post
    Could you all please help me get started on finding the domain and range of these 2 functions

     g(x) = \sqrt{x^2-2x-5}

    Mr F says: Let the domain be {\color{red}(-\infty, ~ -a] \cup [b, ~ +\infty)}. The function is decreasing over the first interval and increasing over the second ......

    and

     g(x) = (25+4x-x^2)^\frac{1}{2}

    Mr F gives a hint: {\color{red} y^2 = 25 + 4x - x^2 \Rightarrow x^2 - 4x + y^2 = 25 \Rightarrow (x - 2)^2 + y^2 = 29} ......

    Thank you

    Update: i can find the domains, but the ranges are giving me trouble
    ..
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    Quote Originally Posted by mr fantastic View Post
    ..
    You've thoroughly confused me
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  4. #4
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    Hello,

    For the first one, the domain will be all x such that x^2-2x-5 \ge 0

    x^2-2x-5=x^2-2x+1-6=(x-1)^2-6

    x^2-2x-5 \ge 0 \Longleftrightarrow (x-1)^2 \ge 6

    Can you solve it ?

    Now the range...

    g(x) = \sqrt{\dots} \ge 0

    Try to continue
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    Quote:
    Originally Posted by silencecloak
    Could you all please help me get started on finding the domain and range of these 2 functions



    Mr F says: Let the domain be . The function is decreasing over the first interval and increasing over the second ......

    and



    Mr F gives a hint: ......

    Thank you

    Update: i can find the domains, but the ranges are giving me trouble



    Quote Originally Posted by silencecloak View Post
    You've thoroughly confused me
    There is a certain pre-requisite knowledge that a student working on questions like these is presumed to have.

    You said you could get the domains. If that's the case, then you should know the value of a and b in . From how you got the domain it should be clear that the value of g starts at zero. If you stop and think about the behaviour of g (and keep in mind my earlier remarks about g decreasing and decreasing) I hope it's clear that the range must be [0, oo). Try drawing a graph of g to see this.

    For the second, the graph of g(x) is a semi-circle. It's the upper half of the circle (x - 2)^2 + y^2 = 29. Draw the semi-circle, get the range by inspection.
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  6. #6
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    Quote Originally Posted by Moo View Post
    Hello,

    For the first one, the domain will be all x such that x^2-2x-5 \ge 0

    x^2-2x-5=x^2-2x+1-6=(x-1)^2-6

    x^2-2x-5 \ge 0 \Longleftrightarrow (x-1)^2 \ge 6

    Can you solve it ?

    Now the range...

    g(x) = \sqrt{\dots} \ge 0



    Try to continue
    am i suppose to put the equation under the square root sign?

    and i was able to find the domains of both, by setting them to 0

    i just dont know the method for the range
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  7. #7
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    Quote Originally Posted by mr fantastic View Post
    Quote:
    Originally Posted by silencecloak
    Could you all please help me get started on finding the domain and range of these 2 functions



    Mr F says: Let the domain be . The function is decreasing over the first interval and increasing over the second ......

    and



    Mr F gives a hint: ......

    Thank you

    Update: i can find the domains, but the ranges are giving me trouble




    There is a certain pre-requisite knowledge that a student working on questions like these is presumed to have.

    You said you could get the domains. If that's the case, then you should know the value of a and b in . From how you got the domain it should be clear that the value of g starts at zero. If you stop and think about the behaviour of g (and keep in mind my earlier remarks about g decreasing and decreasing) I hope it's clear that the range must be [0, oo). Try drawing a graph of g to see this.

    For the second, the graph of g(x) is a semi-circle. It's the upper half of the circle (x - 2)^2 + y^2 = 29. Draw the semi-circle, get the range by inspection.
    I'm studying for a multiple choice timed test, i wont have time to graph and inspect.
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  8. #8
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    Quote Originally Posted by silencecloak View Post
    I'm studying for a multiple choice timed test, i wont have time to graph and inspect.
    If you want to get the right answer I'd advise that you do graph and inspect .....
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