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The domain of a function is the set of input values for which the function is defined. For vector-valued functions, we consider the domain of each component separately. Afterward, we combine these individual domains to find the domain of the entire function.

Think of it like this:

• Each component of a vector-valued function may have different restrictions.
• The most restrictive component determines the domain of the entire function.

For example, in the function \(\mathbf{r}(t)=\langle\sin (t), \ln (t), \sqrt{t}\rangle\), here are the considerations:

- \(\sin(t)\) is unrestricted, so it is defined for all \(t\).
- \(\ln(t)\) is defined only for \(t > 0\).
- \(\sqrt{t}\) is defined for \(t \geq 0\).

Since \(\ln(t)\) has the strictest requirement, the overall domain for \(\mathbf{r}(t)\) is \(t > 0\). This approach ensures that each component of the vector function remains valid within the chosen domain.

The natural logarithm, represented as \(\ln(t)\), is a special function that is only defined for positive values of \(t\). Understanding this function involves a few key aspects:

• It is the inverse of the exponential function \(e^t\).
• The natural logarithm grows very slowly; for example, \(\ln(10)\) is only about 2.3.

Another interesting property is that the natural logarithm approaches \(-\infty\) as \(t\) approaches zero from the positive side.

This means the function cannot take zero or negative numbers as input. Thus, for any expression involving \(\ln(t)\), you must ensure that \(t > 0\).

In vector-valued functions, this requirement can sometimes define the most restrictive domain, as seen in the vector function \(\mathbf{r}(t)=\langle\sin (t), \ln (t), \sqrt{t}\rangle\).

The square root function, usually denoted as \(\sqrt{t}\), is a significant concept in mathematics due to its unique properties and restrictions. Here's what makes the square root so special:

• It is defined only for non-negative values \(t \geq 0\).
• The function maps positive numbers to positive numbers; for instance, \( \sqrt{9} = 3\).

What about zero? The square root of zero is zero, so it's perfectly okay for \(t\) to be zero.

Unlike the natural logarithm, which demands \(t > 0\), the square root only needs \(t\) to be zero or positive, meaning \( t \geq 0\).

When combined with other functions within a vector-valued function, you'll compute the square root requirements with the other domain restrictions
to determine the overall domain. However, in the function \(\mathbf{r}(t)=\langle\sin (t), \ln (t), \sqrt{t}\rangle\), the natural logarithm's restriction of \(t > 0\) already covers what the square root needs,
ensuring the defined range includes all necessary conditions.