Find out the domain of function - three dimensional space, Mathematics

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Find out the domain of each of the following.

 (a) f (x,y) = √ (x+y)

(b) f (x,y) = √x+√y

 (c) f (x,y) = ln (9 - x2 - 9y2)

Solution

(a) In this example we know that we cannot take the square root of a negative number thus this means that we must need,

x + y > 0

 Here is a diagram of the graph of this region.

 

803_Find out the domain of Function - Three Dimensional Space 3.png

(b) This function is not same from the function in the previous part.  Here we must need that,

 x ≥ 0   and  y ≥ 0

 and they actually do need to be separate inequalities. There is one for every square root in the function. 

Here is the diagarm of this region.

808_Find out the domain of Function - Three Dimensional Space 2.png

(c) In this last part we are familiar with that we cannot take the logarithm of a negative number or zero (0). Hence we need to require that,

9 - x2 - 9y2 > 0                  ⇒ x2 /9 + y2 < 1

and on rearranging we see that we require to stay interior to an ellipse for this function.

Here is a diagram of this region.

152_Find out the domain of Function - Three Dimensional Space 1.png


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