Step 1

Given vector-valued function

\(\displaystyle{r}{\left({t}\right)}=\sqrt{{{\left({4}-{t}\right)}^{{{3}}}}}{i}+{t}^{{{2}}}\ {j}-{6}{t}\ {k}\)

Step 2

For the domain of given function

Here

\(\displaystyle{\left({4}-{t}\right)}^{{{2}}}\) is non negative for every real number

\(\displaystyle\Rightarrow\sqrt{{{\left({4}-{t}\right)}^{{{2}}}}}\) is also non negative for every real number

\(\displaystyle\Rightarrow\sqrt{{{\left({4}-{t}\right)}^{{{2}}}}}\) is defined for every real number

And

\(\displaystyle{t}^{{{2}}}\) and \(\displaystyle-{6}{t}\) is polynomial function

The domain of polynomial function is every real number.

So,

Domain of \(\displaystyle{r}{\left({t}\right)}={\mathbb{{{R}}}}\)

or

Domain of \(\displaystyle{r}{\left({t}\right)}={\left(-\infty,\ \infty\right)}\)

Given vector-valued function

\(\displaystyle{r}{\left({t}\right)}=\sqrt{{{\left({4}-{t}\right)}^{{{3}}}}}{i}+{t}^{{{2}}}\ {j}-{6}{t}\ {k}\)

Step 2

For the domain of given function

Here

\(\displaystyle{\left({4}-{t}\right)}^{{{2}}}\) is non negative for every real number

\(\displaystyle\Rightarrow\sqrt{{{\left({4}-{t}\right)}^{{{2}}}}}\) is also non negative for every real number

\(\displaystyle\Rightarrow\sqrt{{{\left({4}-{t}\right)}^{{{2}}}}}\) is defined for every real number

And

\(\displaystyle{t}^{{{2}}}\) and \(\displaystyle-{6}{t}\) is polynomial function

The domain of polynomial function is every real number.

So,

Domain of \(\displaystyle{r}{\left({t}\right)}={\mathbb{{{R}}}}\)

or

Domain of \(\displaystyle{r}{\left({t}\right)}={\left(-\infty,\ \infty\right)}\)