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Find the domain of the vector function. (Enter your answer using interval notation).

Find The Domain Of The Vector Function. Enter Your Answer Using Interval Notation.

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:

\[ ( – \infty , – 9 ) \cup ( – 9 , \infty ) \] 

Image/Mathematical drawings are created in Geogebra.

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