Understanding Domains of Vector-Valued Functions

The video tutorial explains how to determine the domain of vector valued functions by analyzing the intersection of the domains of their components. It provides four examples using different types of functions, such as sine, cosine, logarithm, quadratic, rational, parabola, and square root functions. The tutorial emphasizes the importance of understanding the domains of basic functions and demonstrates how to use set builder and interval notation to express these domains.

What is the domain of a vector-valued function R(t) if its components are X(t), Y(t), and Z(t)?

• The domain of X(t) only

• The domain of Y(t) only

• The intersection of the domains of X(t), Y(t), and Z(t)

• The union of the domains of X(t), Y(t), and Z(t)

What is the significance of the intersection of domains in determining the domain of a vector-valued function?

• It determines the range of the function

• It identifies the common domain where all components are defined

• It is not significant

• It only affects the domain of X(t)

In Example 1, what is the domain of the vector-valued function R(t) with components X(t) = 2sin(t), Y(t) = cos(2t), and Z(t) = ln(t+2)?

• t > -2

• t < 0

• t < -2

• t > 0

In Example 1, which component function restricts the domain of the vector-valued function R(t)?

• Y(t) = cos(2t)

• X(t) = 2sin(t)

• None of the components

• Z(t) = ln(t+2)

For Example 2, which value of t is not included in the domain of the vector-valued function R(t) with components X(t) = 5t^2, Y(t) = 1/t, and Z(t) = 1/(t+3)?

• t = 2

• t = 3

• t = 1

• t = 0

In Example 2, what is the domain of the vector-valued function R(t) using interval notation?

• (-∞, 0) ∪ (0, 3) ∪ (3, ∞)

• (0, ∞)

• (-∞, 3)

• (0, 3)

In Example 3, what is the domain of the vector-valued function R(t) when all component functions have domains of all real numbers?

• All real numbers

• t > 1

• t < 0

• t > 0

In Example 3, what type of graph represents the domain of the vector-valued function R(t)?

• A circle

• A line segment

• A parabola

• The entire t-axis

For Example 4, which value of t is excluded from the domain of the vector-valued function R(t) with components X(t) = 1/4t, Y(t) = 1/t^2, and Z(t) = sqrt(t+4)?

• t = -4

• t = 0

• t = 4

• t = 1

In Example 4, what is the domain of the vector-valued function R(t) using interval notation?

• [-4, 0) ∪ (0, ∞)

• (0, ∞)

• (-∞, 0)

• [0, ∞)