The Domain of a Function of Several Variables

Recall from the Functions of Several Variables page, that the domain of a real-valued function $f$ is defined to by the set of points $(x, y)$ (for two variable functions) or $(x, y, z)$ (for three variable functions) for which $f$ is defined.

Geometrically, we can visualize the domain of a two variable function $f$ as a subset of $\mathbb{R}^2$ that contains all points $(x, y)$ for which $f$ is defined.

Similarly, we can visualize the domains of a three variable function $f$ as a subset of $\mathbb{R}^3$ that contains all points $(x, y, z)$ for which $f$ is defined.

In determining the domain of a function of several variables, it is often important to note which points are NOT contained in the domain. Like mentioned above, sometimes we may restrict the domains forcefully. For example, the function $f(x, y) = x + y$ for $x, y ≥ 0$ represents the portion of the plane $z = x + y$ in the first octant. It is also important to make note of points which make denominators equal to zero, or make arguments of certain functions undefined.

For example, the point $(1, 1)$ is not in the domain of the function $f(x, y) = \frac{1}{x - y}$. In fact, any point $(x, y)$ such that $x = y$ is not contained in the defined for this function.

Another such example is the function $g(x, y) = \sqrt{xy}$. This function is not defined when $xy < 0$. This happens if $x > 0$ and $y < 0$ OR $x < 0$ and $y > 0$.

Of course, we can also graph the domains of two variable functions on an $xy$-plane. Shaded regions and solid lines will be contained within the domain of a two variable function, while non-shaded regions and dotted lines are not contained within the domain of a two variable function.

We will now look at some examples of determining the domains of various functions of several variables.

Example 1

Determine and illustrate the domain of the function $f(x, y) = x^2y^2 + 2x + 2y$.

We note that for any $(x, y) \in \mathbb{R}^2$, $x^2y^2 + 2x + 2y$ is defined. In other words, there is no point $(x, y)$ for which $f(x, y)$ is undefined. Therefore, $D(f) = \mathbb{R}^2$.

Example 2

Determine and illustrate the domain of the function $f(x, y) = \frac{x^2 + y^2}{x - y}$.

We note that both the numerator and denominator of $f$ is defined for all $(x, y) \in \mathbb{R}^2$. However, $x - y \neq 0$, otherwise the denominator would be zero. Therefore the domain of $f$ contains all of $\mathbb{R}^2$ except for the line $y = x$, thus, $D(f) = \mathbb{R}^2 \setminus \{ (x, y) : x = y \}$. The domain of this function is depicted below.

Example 3

Determine and illustrate the domain of the function $f(x, y) = \frac{\sin (xy)}{y - x^2}$.

We note that both the numerator and denominator of $f$ is defined for all $(x, y) \in \mathbb{R}^2$. Once again though, we must note that $y - x^2 \neq 0$. Therefore the domain of $f$ contains all of $\mathbb{R}^2$ except for the parabola $y = x^2$, that is $D(f) = \mathbb{R}^2 \setminus \{ (x, y) : y = x^2 \}$. The domain of this function is depicted below.

Example 4

Determine and illustrate the domain of the function $f(x, y) = \ln (x + y)$.

We note that $x + y > 0$, and so the domain of $f$ contains all points $(x, y)$ such that $y > -x$, in other words, $D(f) = \{ (x, y) : y > -x \}$.

Example 5

Determine and illustrate the domain of the function $f(x, y) = \frac{2\sqrt{x + y}}{x^2 - y}$.

We note that $f$ is defined when $x + y ≥ 0$ so that the numerator is defined, and $x^2 - y \neq 0$ so that the denominator is defined, that is $y ≥ -x$ and $y \neq x^2$, so $D(f) = \{ (x, y) : y ≥ -x \: \mathrm{and} \: y \neq x^2 \}$.

Example 6

Determine and illustrate the domain of the function $f(x, y, z) = \sin (xy) \sqrt{y - z}$.

We note that $f$ is defined when $y - z ≥ 0$, that is $y ≥ z$. Therefore $D(f) = \{ (x, y, z) : y ≥ z \}$.