Functions of two variables

Multivariate functions: Basic notions

Functions of two variables

Intuitive definition of a function of two variables

So far we have limited ourselves to functions of a single variable. These can be viewed as little machines producing a new number from a given number according to a given formula. But you can also think of little machines and formulas that produce a new number from two given numbers. In that case, we speak of a function of two variables.

Recall from the theory Functions that a function from a set $X$ to a set $Y$ assigns a unique element of $Y$ to each element of $X$ . "Functions" in this course are almost always understood to be "real functions". This means $Y=\mathbb R$ , both for real functions of a single variable and for functions of two variables.

For real functions of a single variable, $X$ is a subset of $\mathbb R$ . In the case of a function of two variables, $X$ is a subset of ${\mathbb R}^2$ , the set of all pairs of real numbers, and $Y=\mathbb R$ . Thus, an element of the domain belongs to ${\mathbb R}^2$ and is usually denoted by its coordinates $\rv{x,y}$ .

You may wonder whether there is a notion corresponding to a little machine that produces two numbers (instead of a single number) from two given numbers. In fact, there is: a function from ${\mathbb R}^2$ to ${\mathbb R}^2$ .

Three simple functions of two variables

The area $O$ of a triangle with base $b$ and height $h$ : $O(b,h)=\tfrac{1}{2}\cdot b\cdot h\tiny.$
The distance $s$ covered by an object in uniform motion with velocity $v$ and time $t$ : $s(v,t) = v\cdot t\tiny.$
The milk consumption $x$ as a function of the price $p$ of milk and the average income $m$ per family: $x(m,p)=3\cdot \frac{m^{2.07}}{p^{1.4}}\tiny.$

These examples are written in the form of a function rule, that is a description of the actual machine producing the new number from the given pair $\rv{b,h}$ in the first example and $\rv{v,t}$ in the second example.

In the second example, the function rule is the expression $v\cdot t$ . The dependent variable $s$ is explicitly written down, isolated from the independent variables $v$ and $t$ .

The terminology of functions which we already know is also used for functions of two variables.

Functions of two variables

A relation between three variables $x$ , $y$ , and $z$ , where $x$ and $y$ occur as independent variables, is a function $z=z(x,y)$ if, for any admissible values $x$ and $y$ , there is exactly one value for $z$ corresponding to it. This value $z$ is called the value of the function at the point $\rv{x,y}$ .

The set of all pairs $\rv{x,y}$ of admissible values $x$ and $y$ is called the domain of the function. If $D$ is the domain, we speak of a function on $D$ .

The value of $z$ at a point $\rv{x,y}$ of the domain is called the function value at $\rv{x,y}$ .

The set of all values that the function can assume is called the range of the function.

The graph of a relation between three variables is the set of all points $\rv{x,y,z}$ satisfying the relation. In particular, the graph of a function $f$ of two variables is the set of all points $\rv{x,y,f(x,y)}$ for $\rv{x,y}$ ranging over the points of the domain of $f$ .

The definition of graph is a straightforward generalization of the notion of graph given in the case of a single variable.

Calculate the value of $f(-4,-2)$ for the function $f(p,q)= (2{p}^2+{q})^2-3{p}^2-6{q}$

$f(-4,-2)={}$ $864$

Substituting $p=-4$ and $q=-2$ in the expression $(2{p}^2+{q})^2-3{p}^2-6{q}$ gives $(2\cdot (-4)^2+(-2))^2-3\cdot (-4)^2-6\cdot (-2)\tiny.$ Simplification of this expression gives the answer $864$ .

New example