Finding Domain and Range of a Multivariable Function

Question

for the following function find the domain D and range T and show that for every c ∈ T there exists x,y such that f(x,y) = c. Sketch the Domain.

The following is my own answer but I am unsure if it is correct and was wondering if you could check it for me.

f(x,y) = 8ln(2+x+y²) 

 

Domain: {(x,y)∈ R² : x+y² > -2}

 

Assume Range = R

To determin if this is true, for each c ∈ R we must find a pair (x,y) such that f(x,y) = c 

f(e^c/8-2,0) = 8ln(2+e^c/8 - 2 +0) = 8(c/8) = c , therefore c ∈ range and the range of f(x,y) = R

 

I don't understand what it means when it says sketch the domain?

Answer 1

To graph the set of all points (x,y) where x + y² > -2, first sketch the graph of x + y² = -2.  This is a parabola opening to the left with vertex (-2,0).  Draw the parabola as dotted to indicate that the points on the parabola do not satisfy the inequality.  Next, choose a point not on the parabola and plug it into the inequality.  For example, if we pick (-3,0) and plug in, we get -3 > -2, which is obviously false.  So, (-3,0) is not in the domain of the function.  Since the point (-3,0) is "inside" the parabola and is not in the domain, we can conclude that no points inside the curve are in the domain and that all points "outside" the curve are in the domain.  Shade in the region outside of the dotted parabola.  That region is the graph of the domain.