# Find the domain of the vector functions

• Whatupdoc
In summary, the domain of the vector function r(t) = <ln(6t), sqrt(t+14), 1/sqrt(16-t)> is t in (0,16).
Whatupdoc
Find the domain of the vector functions, r(t), listed below

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

i don't extactly know how to approach this, can someone give me a hint or two

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].

## 1. What is the domain of a vector function?

The domain of a vector function is the set of all possible input values for the function. In other words, it is the range of values that the independent variable can take on.

## 2. How do you find the domain of a vector function?

To find the domain of a vector function, you must first determine the restrictions on the independent variables. This can be done by identifying any denominators, square roots, or logarithms in the function and finding the values that make them undefined. The domain will then be all real numbers except for these restricted values.

## 3. Can the domain of a vector function be infinite?

Yes, the domain of a vector function can be infinite if there are no restrictions on the independent variables. This means that the function can take on any real value as its input.

## 4. What happens if a value falls outside the domain of a vector function?

If a value falls outside the domain of a vector function, the function is considered undefined at that point. This means that the output of the function cannot be determined and the function cannot be evaluated at that value.

## 5. How does the domain of a vector function affect its graph?

The domain of a vector function determines the range of values that will be plotted on the graph. If there are restrictions on the domain, the graph may have breaks or gaps where the function is undefined. Additionally, the domain can also affect the shape and behavior of the graph.

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