This question aims to find the domain of a vector-valued function and the answer should be expressed in an interval notation.
A vector-valued function is a mathematical function that consists of more than one variable that has a range of multi-dimensional vectors. The domain of a vector-valued function is the set of real numbers and its range consists of a vector. Vector or scalar-valued functions can be inserted.
These types of functions play a big role in calculating different curves both in two-dimensional and three-dimensional space.
Acceleration, velocity, displacement, and distance of any variable can be easily found by making vector-valued functions and applying line functions and contours to these functions both in an open and closed field.
Expert Answer
Consider a function:
\[ r ( t ) = \sqrt { 9 – t ^ 2 } i + t ^ 2 j – 5 t k \]
\[ r ( t ) = < 9 – t ^ 2 , t ^ 2 , – 5 t > \]
The set of all real numbers is the domain of rational numbers and the denominator must be a non-zero number. Put the function equal to zero to find the restriction of the domain of rational numbers.
Taking the square on both sides of the equation:
\[ 9 – t ^ 2 = 0 \]
\[ t ^ 2 = 9 \]
\[ t = \pm 3 \]
Domain in interval notation:
\[ ( – \infty , – 3) \cup ( + 3 , \infty ) \]
The component j of the given vector is as follows:
\[ t ^ 2 = 0 \]
Taking square root on both sides of the equation:
\[ t = 0 \]
\[ { t : t \in R } \]
The domain component is all real numbers so it is not restricted to any number.
The component k of the given vector is as follows:
\[ – 5 t = 0 \]
\[ t = 0 \]
The domain of this component is all real numbers so it is not restricted to any number.
Domain in interval notation:
\[ { t : t \in R } \]
Numerical Solution
The domain of a given vector-valued function is $ ( – \infty , – 3) \cup ( + 3 , \infty ) $ for component i and for other components, the domain is all real numbers without any restriction.
Example
\[ f ( t ) = \frac { 7 y } { y + 9 } \]
The set of all real numbers is the domain of rational numbers and the denominator must be a non-zero number. Put denominator equal to zero to find the restriction of the domain of rational numbers.
By setting the denominator equal to zero, we get:
\[ y + 9 = 0 \]
Rearranging the above equation:
\[ y \neq – 9 \]
Hence, – 9 is a number at which the domain becomes restricted. The domain of the given function must lie to the left or right side of this number.
Interval notation:
\[ ( – \infty , – 9 ) \cup ( – 9 , \infty ) \]
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