Understanding the Domain of a Function of Two Variables

The video tutorial explains how to determine the domain of the function f(x, y) = sqrt(x^2 - y^2). It discusses the conditions under which the radicand must be non-negative and solves the inequality x^2 - y^2 >= 0 to find the domain. The tutorial then demonstrates how to graph the inequalities y <= |x| and y >= -|x| on the coordinate plane and verifies the domain using a 3D graph, showing that the domain is in regions B and D.

What is the domain of a function of two variables?

• The set of all possible outputs

• The set of all possible ordered pairs (x, y)

• The set of all possible values of x

• The set of all possible values of y

Why must the radicand x^2 - y^2 be non-negative?

• To ensure the function is complex

• To ensure the function is undefined

• To ensure the function is a real number

• To ensure the function is imaginary

What inequality must be solved to find the domain of f(x, y)?

• x^2 = y^2

• x^2 >= y^2

• x^2 > y^2

• x^2 < y^2

What does the absolute value of y represent when solving the inequality?

• The negative value of y

• The positive value of y

• The non-negative value of y

• The non-positive value of y

How is the inequality y <= |x| represented on the coordinate plane?

• As a parabola

• As a line

• As a V shape

• As a circle

What does the graph of y = -|x| look like?

• A circle

• A parabola

• A line that opens upwards

• A V shape that opens downwards

In which regions are the inequalities y <= |x| and y >= -|x| both true?

• Regions A and B

• Regions A and C

• Regions B and D

• Regions C and D

What shape does the 3D graph of the surface resemble near the x, y plane?

• A cylinder

• A sphere

• A full cone

• A half cone

What axis does the +X-axis point towards after rotation?

• To the right

• Downward

• Upward

• To the left

What is verified by looking down on the x, y plane in the 3D graph?

• The domain of the function

• The maximum value of the function

• The range of the function

• The minimum value of the function