Domain of a Vector Function: Example 1

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Determine the domain of

For many functions, the domain is all of the real numbers. Is that the case here?

No.

In this problem, for which functions do we have to be careful?

We have to be careful with .

What is special about ?

The natural log function is defined only for a positive argument.

What is the consequence here?

Here also, the argument of the natural log must be greater than 0.

Does this mean that t must be greater than 0?

No.

What expression must be positive?

We must have

Solve for the corresponding condition on t.

We have

Draw this interval on the number line on paper. Then check your work by clicking "Next".

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Why do we use parentheses at each end of the interval?

The parentheses indicate that the points at –3 and 3 are not included in the domain.

Now consider . What is special about it?

The square root requires an argument greater than, or equal to, 0.

What is the consequence for

We need

Solve for t.

We get

Show this result on the number line.

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What does the square bracket mean??

It indicates that the point at is included in the domain.

How do these results apply to determining the domain of ?

The domain of must satisfy the most restrictive combination of these inequalities.
Show that interval on the number line.

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Why do we use a square bracket on the left end of the domain and a parenthesis on the right end?

This indicates that the point at the left end is included in the domain and the point at the right end is not.

State this result as an inequality.

The domain of is .

Let's check our work by graphing. Draw a graph of on paper. Then check your work by clicking "Next."

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What do the empty circles mean?

They indicate that the function is undefined at those points.

Draw the graph of . Then click "Next" to check.

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What does the solid circle mean?

It means that the point at is included in the domain of the function.

Combine these two graphs so as to show their common domain.

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Does the graph confirm our result for the domain?

Yes, is the only interval in which both curves exist. For comparison, the domain is shown at the bottom of the graph.

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