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