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Math Help - range of a function of 3 variables ...

  1. #1
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    range of a function of 3 variables ...

    Let g(x,y,z) = ln(25 - x^2 - y^2 - z^2)

    Find the range of g.


    I know this isn't even that complicated I just don't understand how to find range. Thanks for the help!
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  2. #2
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    To find the range, find out what's the maximum and minimum of this function. The natural log function is an increasing function (that is, as x increases from -infinity to +infinity, ln(x) increases). Thus, the maximum that g(x,y,z) can be is ln(25); do you see which this is the case? x^2, y^2, and z^2 are all positive, and they're subtracted from 25. Now, the minimum is a different story; since x, y, and z are unbounded, you can try very large numbers and very small numbers. In fact, it turns out that ln(x) where x<=0 will never work, but ln(x) where x<1 is a negative number. In fact, if you let x^2 + y^2 + z^2 = 24.5, you'll find that g(x,y,z) ~= -0.69. So, let x^2 + y^2 + z^2 = 25-10^-20... that gets you g(x,y,z) ~= -46. Keep trying things; you'll find out that there is no lower bound for g(x,y,z). So your answer is:

    ( -infinity, ln(25) ]
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  3. #3
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    Hello, jlt1209!

    Let g(x,y,z) \:= \:\ln\left(25 - x^2 - y^2 - z^2\right)

    Find the range of g.

    We have: . g(x,y,z) \;=\;\ln\bigg[25 - \left(x^2+y^2+z^2\right)\bigg]

    We see that: . x^2+y^2+z^2 \:\geq \:0\quad\hdots a sum of squares is always nonnegative,

    . . also that: . 25 - (x^2+y^2+z^2) \:>\:0 \quad\Rightarrow\quad x^2+y^2+z^2 \:< \:25

    Hence: . 0 \:\leq x^2+y^2+z^2 \:<\:25


    If x^2+y^2+z^2 \:=\:0,\;\;g(x,y,z) \:=\:\ln(25)

    If x^2+y^2+z^2\: \to\: 25,\;\;g(x,y,z) \:\to \:\text{-}\infty


    Therefore, the range is: . \bigg(\text{-}\infty,\;\ln25\bigg]

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