2.1 Domains and ranges of multivariable functions

Function Types

Multivariable and Vector-Valued Functions

A multivariable function takes input from two or more variables and maps them to a single output value. A vector-valued function, by contrast, returns a vector as its output rather than a single number.

The components of a vector-valued output are themselves functions of the input variable(s). For instance, $\vec{r}(t) = \langle \cos(t), \sin(t), t \rangle$ takes a single input $t$ and produces a 3D vector. Each component traces out its own behavior as $t$ changes, and together they describe a helix in space.

Note that a function can be both multivariable and vector-valued. These functions show up when modeling systems like fluid flow fields or electromagnetic fields, where multiple spatial inputs determine a vector quantity at each point.

Scalar-Valued Functions

A scalar-valued function maps one or more input variables to a single real number (a scalar). For example, $f(x, y) = x^2 + y^2$ takes the pair $(x, y)$ and returns one number.

When a scalar-valued function has two inputs, you can graph it as a surface in 3D space, with the two inputs on the horizontal axes and the output on the vertical axis. These functions appear constantly in optimization (finding maxima and minima) and in modeling physical quantities like temperature or pressure that have a single value at each point in space.

Domain and Range

Domain of Multivariable Functions

The domain is the set of all input values where the function is defined. For a function $f(x, y)$, the domain is a region in the $xy$-plane (a subset of $\mathbb{R}^2$).

Some functions have unrestricted domains. $f(x, y) = x^2 + y^2$ accepts any real values of $x$ and $y$, so its domain is all of $\mathbb{R}^2$.

Other functions have natural restrictions imposed by the formula itself. To find the domain, look for the usual trouble spots:

• Square roots (or even roots): the expression under the radical must be $\geq 0$
• Denominators: any expression in a denominator cannot equal zero
• Logarithms: the argument must be $> 0$

For example, $g(x, y) = \frac{1}{\sqrt{9 - x^2 - y^2}}$ requires $9 - x^2 - y^2 > 0$, so the domain is the open disk $x^2 + y^2 < 9$. The boundary circle is excluded because the denominator would be zero there.

Physical context can also restrict the domain. A function modeling the height of a terrain might only be defined for coordinates within a certain region.

Range and Codomain

The range is the set of all output values the function actually produces when you feed in every point from the domain.

• For scalar-valued functions, the range is a subset of $\mathbb{R}$.
• For vector-valued functions, the range is a subset of the output space (e.g., $\mathbb{R}^3$).

The codomain is the target space you declare for the function's outputs. It can be larger than the range. For example, you might say $f : \mathbb{R}^2 \to \mathbb{R}$, making $\mathbb{R}$ the codomain, but if $f(x,y) = x^2 + y^2$, the range is only $[0, \infty)$ since the function never produces negative values.

To find the range, apply the function to all inputs in the domain and determine which output values are achievable.

Coordinate Systems

Cartesian Coordinates

Cartesian coordinates $(x, y, z)$ locate points in 3D space using three mutually perpendicular axes. Each coordinate gives the signed distance from the origin along its axis.

This is the default system for working with multivariable functions. A function like $f(x, y) = x^2 - y^2$ translates directly into a surface over the $xy$-plane. Cartesian coordinates are especially natural when the axes carry distinct physical meanings (e.g., two spatial dimensions and one for time) or when the relationships involved are linear.

Projection and Coordinate Planes

A projection maps points from a higher-dimensional space down to a lower-dimensional subspace by "dropping" one or more coordinates.

The three coordinate planes in $\mathbb{R}^3$ are the standard projections:

• The $xy$-plane: set $z = 0$
• The $xz$-plane: set $y = 0$
• The $yz$-plane: set $x = 0$

These planes are useful for analyzing multivariable functions through their traces. A trace is the curve you get by slicing a surface with a plane parallel to one of the coordinate planes. For example, setting $y = 0$ in $f(x, y) = x^2 - y^2$ gives the trace $z = x^2$, a parabola in the $xz$-plane. Examining several traces builds intuition for the shape of the full surface without needing to visualize all three dimensions at once.