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# Range of a Multivariable Function

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Hey guys,

How do I find the range of f(x,y)=sqrt(9-x^2-y^2)?

Thanks for the help!

mathmeme  Aug 4, 2017
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### 3+0 Answers

#1
+91259
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How do I find the range of f(x,y)=sqrt(9-x^2-y^2)

$$\sqrt{9-x^2-y^2}\\ =\sqrt{9-(x^2+y^2)}\\ \text{So for real solutions}\\ x^2+y^2\le9\\ -3\le x \le3\qquad and \qquad -3\le y\le3\\ \text{The range is } -3\le y\le 3$$

Melody  Aug 4, 2017
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f (x,y)  = √ [ 9 - x ^2 - y^2 ]  = √ [ 9 - ( x ^2 + y^2) ]

We have a 3D surface here

Let  f(x,y)  = z

In real terms, the minimum for z, 0, will be achieved when x^2 + y^2   = 9

And the maximum for z, 3, will be achieved when   x , y   = 0

And the range of z will be      0 ≤ z ≤ 3

CPhill  Aug 4, 2017
edited by CPhill  Aug 4, 2017
#3
+91259
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Thanks Chris,

I guess it was the the restriction on z scores that were wanted.   :/

Melody  Aug 7, 2017

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