91Ó°ÊÓ

The domain of a vector function is the set of all possible input values, usually represented by the variable, for which the function components are well-defined. For a vector function such as \( \mathbf{r}(t)=\langle \ln(11t), \sqrt{t+10}, \frac{1}{\sqrt{12-t}} \rangle \), we need to find out where each component is defined or undefined.

Consider the logarithmic function \( \ln(11t) \). Logarithmic functions are only defined for positive arguments. Thus, for \( \ln(11t) \) to be defined, \( 11t > 0 \) or \( t > 0 \).

Next, the square root function \( \sqrt{t+10} \) is defined for non-negative arguments, meaning \( t+10 \geq 0 \), simplifying to \( t \geq -10 \).

Finally, the expression \( \frac{1}{\sqrt{12-t}} \) requires \( \sqrt{12-t} eq 0 \) and \( 12-t > 0 \), which leads to \( t < 12 \) and makes sure \( t eq 12 \).

By evaluating these restrictions, we can determine the precise domain of the function.

Interval notation is a concise way to express the domain using brackets and parentheses. Round brackets \(( )\) indicate that a boundary is not included, while square brackets \([ ]\) mean it is included.

When we have determined a domain, we express it in interval notation. For instance, if \( t \) must be greater than 0 and less than 12, we represent this as \((0, 12)\).

This notation shows clearly that the values of \( t \) are any real number greater than 0 but less than 12, not inclusive of the endpoints. This format is widely used in mathematics for its simplicity and clarity, especially when dealing with continuous domains.

Restrictions arise from the nature of mathematical functions where certain operations are not valid for all numbers. In our example, different components have different limitations which restrict \( t \).

- **Logarithmic Function:** The expression \( \ln(11t) \) is only defined when the argument is positive, restricting \( t > 0 \).

- **Square Root Function:** For \( \sqrt{t+10} \) to be real and defined, the value under the root must be non-negative, making \( t \geq -10 \) the condition for validity.

- **Division by Zero:** \( \frac{1}{\sqrt{12-t}} \) includes both a square root and a fraction, and hence, the radical should be positive (not zero), constraining \( t < 12 \).

After identifying all these restrictions, we determine their overlap by finding the most restrictive set of conditions which is \( 0 < t < 12 \). Applying these conditions ensures each function within the vector is defined and meaningful throughout the chosen domain.