# Find the domain of the vector functions

Find the domain of the vector functions, r(t), listed below

a.) r(t) = <ln(6t), sqrt(t+14), 1/sqrt(16-t)>

i dont extactly know how to approach this, can someone give me a hint or two

## Answers and Replies

The domain of a function $$f(x)$$ basically means the set of all $$x$$ for which the function yields valid results. In your case, I presume all vector components must be real numbers. What does that mean for the values that $$t$$ is allowed to have?

$$t \geq -14$$
$$t \leq 16$$

right? so the domain should be from [-14,16]

You're on the right track, but what about the ln(6t) function?

t cannot equal to 0 for ln(6t). so should it be [-14,0) U (0, 16]

If ln(6t) is to yield a real number, t must be greater than 0. A negative value for t won't work! So one of the domains you posted is almost right - the one holding positive values. The reason that the interval isn't exactly correct is because t = 16 isn't allowed (take a look at the third term and see what happens when t=16). In other words: t is in (0,16) instead of (0,16].