Labware - MA35 Multivariable Calculus - Three Variable Calculus
 MA35 Labs 3 » Three Variable Calculus Contents3.1 Functions of Three Variables 3.1.1 Linear Functions 3.1.2 Domain, Range & Function Graphs 3.1.5 Continuity 3.4 Integration Search

Domain, Range & Function Graphs (Page: 1 | 2 )

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The domain of a function of three variables is a subset of coordinate 3-space \{ (x,y,z) | x, y \mbox{and} z ∈ {R} \}.

The range of a real-valued function f is the collection of all real numbers f(x,y,z) where (x,y,z) is in the domain of f.

The graph of a function of three variables is the collection of points (x,y,z,f(x,y,z)) in 4-space where (x,y,z) is in the domain of f.

Demos

 Domain and Range This demonstration shows the three-dimensional domain and the (one-dimensional) range of a function f(x,y,z). By default, the domain is set to -1 {leq} x {leq} 1, -1 {leq} y {leq} 1, -1 {leq} z {leq} 1 and the function is set to f(x,y) = x2 - y2 + z2. In the window labeled Domain and Range, you can choose both the center and radius of a red spherical domain using the white and red hotspots, respectively. The magenta line segment along the z-axis in this window shows the range of f(x,y,z) of this red domain.

Exercises

• 1. What is the range of the function f(x,y,z) = ax2 + cy2 + ez2? (The answer will depend on the constants a, c, and e.)
• 2. What is the range of the function f(x,y,z) = -x4 + 2x2 - y4 + 2y2 - z4 + 2z2 ?
• 3. In the second demo, whereas every point p in the cube domain -1 {leq} x, y, z {leq} 1 gets assigned a color according to its function value f(p), it is only possible to view the colors that appear on the faces of the cube. When is the range of the colors on the faces the same as the range on the inside of the cube?
• 4. More generally, what condition has to be satisfied for the range of function values f(p) over a three-dimensional domain to be the same as the range over its boundary?
• 5. What is the range of the function g(x,y,z) = ax2 + 2bxy + c2 + 2dyz + e2z2 + 2fzx? (The answer will depend on the constants a, b, c, d, e, and f.)