Find the domain of the vector-valued function r(t)=\sqrt{(4-t)^{2}}i+t^{2]j-6t\ k

skomminbv 2021-11-28 Answered
Find the domain of the vector-valued function
\(\displaystyle{r}{\left({t}\right)}=\sqrt{{{\left({4}-{t}\right)}^{{{2}}}}}{i}+{t}^{{{2}}}{j}-{6}{t}\ {k}\)

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Expert Answer

Huses1969
Answered 2021-11-29 Author has 6937 answers
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 \(\displaystyle{r}{\left({t}\right)}={\left(-\infty,\ \infty\right)}\)
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