Functions and relations

Multivariate functions: Basic notions

Functions and relations

In the natural sciences mathematical models are often used to describe relations between quantities so as to understand processes or to make predictions.

Mathematically a relation between three variables is a subset of ${\mathbb R}^3$ , but in practice it is often given by just writing down an equation.

If $f$ is a function of two variables, then the graph of $f$ is a relation between the three variables; it is the subset of ${\mathbb R}^3$ consisting of all points $\rv{x,y,z}$ satisfying the equation $z=f(x,y)$ .

There is no limit to the number of equations that can be used to describe a relation. For instance, $\eqs{x&=&\e^{t}\cr y&=&\e^{2t}\cr}$ is a pair of equations defining a relation between the three variables $x$ , $y$ , and $t$ .

Relations that are graphs of functions

A relation between three variables $x$ , $y$ , and $z$ is the graph of a function $z=z(x,y)$ if, for any pair of admissible values for $x$ and $y$ , there is exactly one value of $z$ such that $\rv{x,y,z}$ belongs to the relation.

When we are given an equation defining a relation between three variables, we can try to rewrite it in a form where one of the variables, say $v$ , appears on the left hand side of the equation and does not appear on the right hand side. The act of creating such an equation of this form, that is, $v = \text{a formula without }v\tiny,$ is called isolating the variable $v$ .

If the relation defines a function, we call that function implicitly defined by the relation.

Two simple relations

Many relations are not as explicit as a function. For example, the formula for a lens with a focal length $f$ is $\frac{1}{b}+\frac{1}{v}=\frac{1}{f}\tiny,$ where $v$ is the object distance and $b$ the image distance.
Another example of a relation between three variables is the sphere with center $\rv{0,0,0}$ and radius $1$ . It is determined by the equation $x^2+y^2+z^2=1$ .

Sometimes, such a relation is the graph of a function, but not always. In the first example, $f$ is a function of $b$ and $v$ . By isolating the variable $f$ , we find the functional rule $f(b,v)=\frac{1}{\dfrac{1}{b}+\dfrac{1}{v}}\tiny,$ which can, of course, be simplified to $\frac{b\cdot v}{b+v}$ .

In the example of the sphere, $z$ cannot be written as a function of $x$ and $y$ . The reason is that for many points $\rv{x,y}$ , there are two values of $z$ such that $\rv{x,y,z}$ satisfies the equation. We can describe the relation as the union of the graphs of the two functions $z_1$ and $z_2$ , where $z_1(x,y)= \sqrt{1-x^2-y^2}\quad\text{and}\quad z_2(x,y)= -\sqrt{1-x^2-y^2}\tiny.$

In the equation $y=\frac{11x+3z}{-9x+5z}$ you see right away that $y$ is a function of $x$ and $z$ .

But $z$ is also a function of $x$ and $y$ . What is the corresponding function rule? In other words, express $z$ in terms of $x$ and $y$ and enter your answer in the form $z=\text{ an expression in } x\text{ and }y\tiny.$

$z={{\left(9\cdot y+11\right)\cdot x}\over{5\cdot y-3}}$

This can be found by isolating the variable $z$ in the given equation $y=\frac{11x+3z}{-9x+5z}$ :
$\begin{array}{rcl} (-9x+5z)\cdot y &=& 11x+3z\\ &&\phantom{xyz} \color{blue}{\text{denominator cleared}}\\ -9x\cdot y+5y\cdot z-3z&=&11x\\ &&\phantom{xyz} \color{blue}{\text{terms with } z \text{ to the left}}\\ 5y\cdot z-3z&=&11x+9x\cdot y\\ &&\phantom{xyz} \color{blue}{\text{terms without } z \text{ to the right}}\\ (5y-3)\cdot z&=&11x+9x\cdot y\\ &&\phantom{xyz} \color{blue}{\text{terms with } z \text{ collected}}\\ z&=&\frac{11x+9x\cdot y}{5y-3}\\ &&\phantom{xyz} \color{blue}{\text{divided by the coefficient of } z}\\ \end {array}$ We bring $x$ outside the brackets and find the following function rule of $z$ as an answer: $z={{\left(9\cdot y+11\right)\cdot x}\over{5\cdot y-3}}\tiny.$

New example