How to find the domain of a vector function?
Understand the Problem
The question is asking for the method to determine the domain of a vector function, which involves identifying the set of input values for which the function is defined.
Answer
The domain of a vector function is the intersection of the domains of its individual components.
Answer for screen readers
The domain of a vector function is expressed as the intersection of the domains of its individual components.
Steps to Solve
- Identify the vector function components
A vector function can typically be expressed in the form $\mathbf{r}(t) = \begin{pmatrix} x(t) \ y(t) \ z(t) \end{pmatrix}$. Identify each component function, such as $x(t)$, $y(t)$, and $z(t)$.
- Find the domain of each component
For each component function, determine the values of $t$ for which the function is defined. Look for restrictions like division by zero or square roots of negative numbers.
- Combine the domains
The overall domain of the vector function is the intersection of the domains of its component functions. This means you take the set of values for which all components are simultaneously defined.
- Express the domain
Finally, write the domain in interval notation, set notation, or any respective form that clearly indicates the valid input values for the vector function.
The domain of a vector function is expressed as the intersection of the domains of its individual components.
More Information
Understanding the domain of a vector function is essential as it helps define the valid inputs for modeling physical phenomena, determining paths in space, and solving equations involving the vector function.
Tips
- Ignoring restrictions from component functions: Make sure to check each function for restrictions.
- Confusing union with intersection: Remember to find the intersection of domains, not the union.
