Given the function,
[tex]t(v) = \sqrt{ {v}^{2} + 2v - 8 } [/tex]
We are aware that square root function is defined for only non negative values.
So,
[tex]\to {v}^{2} + 2v - 8 \geqslant 0[/tex]
[tex] \to{v}^{2} + 4v - 2v- 8 \geqslant 0[/tex]
[tex]\to {v}(v + 4) - 2(v + 4) \geqslant 0[/tex]
[tex]\to(v + 4)(v - 2) \geqslant 0[/tex]
Therefore,
[tex]\boxed{\red{v\in (-\infty, -4] \cup [2,\infty})} [/tex]
Since this is the possible input values of v for which the given function is defined, it is the domain of the function.
