Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

Chapter 3

Functions of Two Variables

1 Functions of Two Variables

A function f of the two variables x and y is a rule that assigns to each

ordered pair ()x,y of real numbers in some set one and only one real number

denoted byf()xy,.

The Domain of a Function of Two Variables

The domain of the function f(xy, ) is the set of all ordered pairs ()x,y of real

numbers for which f(xy, )can be evaluated.

Example 1

For f()xy,3=+x y^2. Find a. f(2, 3), b.f(2, 2)?

Solution

a. f()2, 3 =×+ =3 2 3^215

b. (^) () ()
2
f 2, 2 =3 2×+ = 2 8
Example 2
Forf()xy e,=+x lny. Find a. the domain of f. b. f(0,1)
Solution
a. For x to be defined, we must have. For lny to be defined, we must
have y>0. So the domain is
x≥ 0
{(, )/xy x≥>0andy^0 }^
b. fe e(0,1)=+=+=^00 ln1 0 1
Example 3
Suppose ()

352

,

x y

fxy

x y

+

=

−

. a. Find the domain of f. b. Computef(1, 2− ).

Example 4
A pharmacy sells two brands of aspirin. Brand A sells for $1.25 per bottle and Brand
B sells for $1.50 per bottle.
a. What is the revenue function for aspirin?
b. What is the revenue for aspirin if 100 bottles of Brand A and 150 bottles B
are sold?
Solution
a. Let x= the number of bottles of Brand A sold and y= the number of
bottles of Brand B sold.Then, the revenue function is
R()xy, =+1.25x1.50y

b. R()100,150 =×+×=+=1.25 100 1.50 150 125 225 350