describe the domain of a function $f(x, y)$

Question

Describe the domain of the function $f(x,y) = \ln(7 - x - y)$. I have the answer narrowed down but I am not sure if it would be $\{(x,y) \mid y ≤ -x + 7\}$ or $\{(x,y) \mid y < -x + 7\}$ please help me.

Answer 1

Hint: is $\ln 0$ defined? This should help you determine which of your two answers is correct.

Answer 2

The domain is the locus of points $D=\{(x,y)\in\mathbb R^2 | 7-x-y>0\}$. You just need to work on the inequality, now. Of course you should know the domain of $\log$...

Answer 3

There are many possible descriptions, such as the set of all $(x,y)$ such that $x+y\lt 7$. That to me is the nicest, because it is quite symmetrical.

But it can be rewritten, as you did, as the set of all $(x,y)$ such that $y\lt 7-x$. You don't want $y\le 7-x$, since $\ln$ is not defined at $0$.

Answer 4

Nice job: on the inequality part of things; the only confusion you seem to have is with respect to whether to include the case $y = x-7$. But note that $$f(x, y) = f(x, x - 7) = \ln(7 - x - y) = \ln \left[7 - x -(7 - x)\right] = \ln 0$$ but $\;\ln (0)\;$ is not defined, hence $y = x - 7$ cannot be included domain!

So we want the domain to be one with the strict inequality: $$\{(x,y) | y < -x + 7\}$$