2. Multivariable Functions

Function of Several Variables

Definition

A function $f$ : $R^n\to R$ associates to each point $(x_1,x_2,\dots ,x_n)$ in its domain exactly one number $f(x_1,\dots ,x_n)$.

Formula Example w/ One Variable

Falling Object: $H(t)=200-5t^2$

$H(5)=200-\frac{1}{2}\cdot 10\cdot 5^2=75$

Formula Example w/ Several Variables

Falling Object: $H(h,t,g)=h-\frac{1}{2}g{t}^2$

$H(200,5,10)=200-\frac{1}{2}\cdot 10\cdot 5^2=75$

Maximal Domain

Definition

The maximal domain of $f(x_1,\dots ,x_n)$ consists of all points $(x_1,x_2,\dots ,x_n)\in R^n$ for which the formula for $f$ produces a real number.

Example

Consider the function $f$ given by

$f(x_1,x_2,x_3,x_4)=\frac{\sqrt{x_1+1}-\ln (4-x_2^2-x_3^2)}{x_4^2-2}$

then the maximal domain consists of all points $(x_1,x_2,x_3,x_4)$ such that

• $x_1\ge -1$
• $x_2^2+x_3^2<4$
• $x_4\ne \sqrt{2}$ and $x_4\ne -\sqrt{2}$

Range

Definition

The range $R$ of a function $f$ : $D\to R$ for $D\subset R^n$ is the set of all possible values of $f(x_1,\dots ,x_n)$ for $(x_1,\dots ,x_n)\in D$:

$R=f(x_1,\dots ,x_n)|(x_1,\dots ,x_n)\in D$