Functions of several variables

In one-variable calculus, we studied functions like $f(x)=x^2$ with one input and one output. When studying curves, we looked at vector-valued functions like ${\bf r}(t) = \langle \cos(t), \sin(t) \rangle$ with one input and several outputs, or equivalently with a vector as the output. Next we are going to study functions like $f(x,y) = x^2 + y^2$ that have several inputs and one output. We call these functions of several variables.

You find functions of several variables throughout science, mathematics, and day-to-day life. On a hot summer day, the heat index is a function of temperature and humidity. On a cold winter day, the wind chill factor is a function of temperature and wind speed. On a Earth, elevation is a function of latitude and longitude. Your blood pressure is a function of your age and how much exercise you get (and other factors, too). The fraction of Democratic (or Republican) voters in a county can be modeled by a complicated function of population density, average income, average education level, concentration of minorities, and rate of church attendance. It's not much of an exaggeration to say that all quantities in the real world are functions of several variables.

For simplicity, we're going to start with functions of just two variables. We'll usually call the variables $x$ and $y$, and call the function $f(x,y)$. The domain is the set of pairs $(x,y)$ that are valid inputs, and the range is the set of output values. The domain is a subset of the $x$-$y$ plane, and the range is a subset of the real line.

Example: Find the domain and range of the function $f(x,y) = \sqrt{1-x^2-y^2}$.

Solution: The domain is the unit disk $x^2 + y^2 \le 1$, since the square root only makes sense if $1-x^2-y^2 \ge 0$. The range is the interval $[0,1]$.