Question

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ g(x, y)=\ln (4-x-y) $$

Short answer

The domain of the function \(g(x, y) = \ln (4 - x - y)\) corresponds to a region in the xy-plane which includes all points below the line \(y = 4 - x\).

Step-by-step solution

Setting up the inequality

We want values of x and y such that \(4 - x - y > 0\). This inequality gives the domain of the function.

Manipulating the inequality

Reforming the inequality makes it easier to grasp. It can be written as \(y < 4 - x\). This means that the function is defined for values of y that are less than \(4 - x\).

Interpreting the inequality as a region

The inequality \(y < 4 - x\) describes a half plane in the xy-plane. The line \(y = 4 - x\) is the boundary of the region, and y-values below this line are included in the region. So, the function is defined for all points below the line \(y = 4 - x\).