The earlier study of Quadric Surface will serve us as we graph a function of two independent variables.

When we write z = f(x,y) we are thinking of an equation for our function. When we think of the domain and range of the function we might better write f(x,y) = z or ( and this will be the case in advanced mathematic's courses ) f:(x,y)--> z. An ordered pair (x,y) is being mapped to a real number z . When graphing, "look for" the (x,y) pairs in a region in the xy-plane and the real numbers for z on the vertical real number line labelled z . You should learn to be able to shade the domain and range in a graph of a function of two independent variables. [See 13.1 top pg. 886]: graph of a function of two variables ] The surface is made up of the graph of ordered triplets ( x,y,z ) which are the result of the pairing of the ( x,y ) and the z , as follows: ( (x,y) , z ) .

We might, for example, work with the equation f(xy) = 4x + xy or , as an equivalent without the use of functional notation, z = 4x + xy. If asked for f(2,3) we are considering the "image" of the ordered pair or "point" (2,3) under the function f and it is the real number 14 ( some will refer to it as the output or

result or the answer ) . Sometimes it is helpful to think of the rule that the equation is "giving" us. In this case it would be to quadruple x and add the product of x times y.

Look again at the graphing aspects of a Quadric Surface which is now being used as the surface graph for a function of two independent variables: [ See 13.1 top pg. 886]

Look at the following solved problems in the Exercises at the end of 13.1 : 5,13,17,31,33,35 Then do the following unsolved as homework : 6,14,18,32,34,36