# Thread: range of a function of 3 variables ...

1. ## 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!

2. 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) ]

3. Hello, jlt1209!

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

Find the range of $\displaystyle g.$

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

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

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

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

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

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

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