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

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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Huses1969
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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