Find the domain of f(x, y) = ln(x + y − 1).
Think about your answer, then reveal below.
Model answer: The domain is the set of (x, y) where x + y − 1 > 0, i.e., x + y > 1. This is the open half-plane above the line x + y = 1.
The natural logarithm requires a strictly positive argument, so we need x + y − 1 > 0, or equivalently x + y > 1. In the plane, x + y = 1 is a line with slope −1 passing through (1, 0) and (0, 1). The domain is the open half-plane on the side where x + y exceeds 1 (upper-right region). Note the domain is open because it excludes the boundary line itself.