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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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