Chapter 2At the end of this section, you should be able to Find
the domain and range of a function of 2variables or more, Perform
operations on functions, and Sketch the graph of a function of 2
variables. Cartesian Coordinate System Distance between two points(
){ }2, , R x y x y R = e( ) ( )21 1 2 2, , , J x y P x y R e( ), d
J P = JP =( ) ( )2 21 2 1 2x x y y + The set of all ordered
n-tuples of real numbers is called the.NOTATION: Point:nR( )1 2 3,
, ,...,nx x x x n-dimensional number space Distance between two
points( ){ }1 2, ,...,nn iR x x x x R = e( )1 2, ,..., ,nJ x x x(
), d J P = JP =( ) ( ) ( )2 2 21 1 2 2...n nx y x y x y + + + ( )1
2, ,...,nnP y y y R e3 n =Ais a set of ordered pairs, such that no
two distinct ordered pairs have the same first element.A is a set
of ordered pairs of the form , such that no two distinct ordered
pairs have the same first element.NOTE:( ), P wnP R ew R e(), f P w
=If , then DOMAIN: RANGE: ( )f P w =the set of all admissible
points Pthe set of all resulting values of w1.2. ( ),xf x yy=( ),
ln ln f x y x y = +DOMAIN:( ){ }, 0 x y y =DOMAIN: ( ){ }, , 0 x y
x y >xyxyf3. ( )2 2 2,, 25 f x y z x y z = DOMAIN:( ){ }2 2 2,,
25 x y z x y z + + sf4.( )( )2 2,, ln f x y z z x y = DOMAIN:( ){
}2 2,, 0 x y z z x y >f5. ( ),, f x y z y xz =DOMAIN:( ){ },, 0
x y z xz >f( ), cosyf x y e x =( )( )2,, ln g x y z xy z = 1.2.
, 03ft| | |\ .11,1,4g| | |\ .0cos3et=12=112= 21ln 14| |= |\
.15ln16=ln15 ln16 = The composite function( )( ) ( )( )f g x f g x
= ( )( )f g x DOMAIN:( ){ }g fx x D g x D e e Let and .( )( ) ( )(
)f g x f g x = ( )sin ft t =DOMAIN:( )1 g x x = sin 1 x = ( ){ }g
fx x D g x D e e( ){ }gx x D g x R = e e{ }gx x D = egD =| )1, =
+If is a function of a single variable and is a function of
variables, the composite functionis the function ofvariables
defined by( )( ) ( ) ( )1 2 1 2, ,..., , ,...,n nf g x x x f g x x
x = ( )f g ( )(){ }1 2, ,...,n g fP x x x P D g P D e efgnn( )tft e
=( )( )2,, ln g x y z xy z = f g DOMAIN:( ) (){ },,g fP x y z P D g
P D e e( ) (){ },,gP x y z P D g P R = e e( ){ },,gP x y z P D =
egD =( ){ }2,, 0 x y z xy z = >( )tft e =( )( )2,, ln g x y z xy
z = f g ( )( ),, f g x y z ( )( ),, f g x y z =( )2ln xy ze=2xy z =
1.( )2 21,1g x yx y=+ 3.( )( ), sinxf x y ye=Optional:Range2.( )2
2,, 9 h x y z x y = 1.( )( ), sinxf x y ye=Domain:Range:2R| |1,1
xy( )2 21,1g x yx y=+ 2.Domain:Range:( ){ }2 2, 1 x y x y + >(
)0, +xy3.( )2 2,, 9 h x y z x y = Domain:Range:( ){ }2 2,, 9 x y z
x y + s| |0, 3xyIf is a function of variables, then the graph ofis
the set of all pointsin for whichAnd.( )1 2, ,..., ,nx x x wfnf1 nR
+( )1 2, ,...,n fx x x D e( )1 2, ,...,nw f x x x =( )2 2, 25 f x y
x y = xyThe set of all points inwherehasa constant valueis called
alevel curve of f.The set of all points in wherehasa constant
valueis called alevel surface of f .2R( ), f x yk3R( ),, f x y zcA
set of level curves or level surfaces obtained by considering
different values of is called a contour map.k1.( )2,yf x yx=( ), f
x y k =2ykx =2y k x =That is, the level curve off is a parabola if
0 , k =line if0. k =Range: R( )2,yf x y kx= =2y k x =0 . y = Let0 k
=2. y x = 1 k =2. y x = 1 k = xy( )2,yf x yx=A sketch of the graph
off is shown below:xyxy1 k =1 k = 0 k =( )2,yf x yx=A sketch of the
graph off is shown below:xy1 k =1 k = 0 k =xy2.( )2 2, g x y y x =
( ), g x y k =2 2y x k =That is, the level curve offis a vertical
hyperbolaif 0 , k >is a horizontal hyperbola if 0 , k
