Goal: recognize a function of two variables and identify its domain and range
Function of two variables
The definition of a function with two variables is similar to that of the function with one variable. The only difference is that we map a pair of two variables to another variable instead of mapping one variable to another variable.
Definition
A function of two variables z = f(x,y) maps each ordered pair (x,y) in a subset D of the real plane R² to a unique real number z. The set D is called the domain of the function. The range of f is the set of all real numbers z that has at least one ordered pair (x,y) 𝜺 D such that f(x,y) = z as shown in the following figure:
Solution
a. This is an example of a linear function with two variables. There is no pair of variables (x, y) for which the function is not defined. Therefore the domain of the function is R².
To determine the range of the function, let's determine the set of reals z for which f(x,y) = z. We have 3x +5y + 2 = z. Let's solve this equation by choosing x = 0. We have 5y + 2 = z. y = z-2/5. The pair (0, z-2/5) is a solution of the equation 3x + 5y + 2 = z for any value of z. The range of the function is R.
b. For the function g to have a real value we need 9 - x² - y²≥ 0 or - x² - y²≥ -9 x² + y² ≤ 9
The domain D is defined as follow: D = {(x,y)ε R²/ x² + y² ≤ 9}.The graph of this set of points is described as a disk of radius 3. The graph includes the boundary as shown below.
To determine the range of the function, we have to find out the set of reals for which g(x,y) = z. The domain is made of circles starting from (0, 0) and ending at the boundary circle defined by x² + y² = 9. Let's find z for (0,0) i.e a point of the domain starting at the origin. We have g(0,0) = z.
g(0,0) = ⎷9-(0)²-(0)² = ⎷9 = 3. Let's take a point of the boundary circle i.e (0,3). We have g(0,3) = ⎷9-(0)² - (3)⁰ = 0. The range is [0,3].
Practice
No comments:
Post a Comment