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](https://www.expertsmind.com/CMSImages/803_Find%20out%20the%20domain%20of%20Function%20-%20Three%20Dimensional%20Space%203.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](https://www.expertsmind.com/CMSImages/808_Find%20out%20the%20domain%20of%20Function%20-%20Three%20Dimensional%20Space%202.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](https://www.expertsmind.com/CMSImages/152_Find%20out%20the%20domain%20of%20Function%20-%20Three%20Dimensional%20Space%201.png)