Finding the Domain of Multivariable Functions

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We’ve talked about finding the domains of quotient functions and functions involving square roots and other functions with limited domains, and it turns out this is pretty much all we need to know to find the domains of multivariable functions.

As an example, consider the function f(x,y)=\frac{1}{x-y}. This function is defined wherever its denominator is nonzero, or in other words, whenever x\neq y. Geometrically, the domain is the entire xy plane with the diagonal line y=x removed. We would write this in set notation as \{(x, y)\colon x\neq y\}.

What about g(x, y)=\ln (x^2+y^2)? We know that the natural logarithm is defined only for positive values of its argument, so in this case we must have x^2+y^2 > 0. The only time when x^2+y^2=0 is when both x and y are zero, so the domain in this case is the entire xy plane minus the origin, or \{(x,y)\colon x\neq 0\text{ and } y\neq 0\}.

Let’s look at one more example: h(x,y) = \sqrt{x-y}. Here we need the argument of the square root to be nonnegative, so we must have x-y\geq 0, or y\leq x. This set is the entire region to the right of the line y=x in the xy plane, and in set notation \{(x,y)\colon y\leq x\}.