g(t) = square root of 3-t - square root of 2+t. Find the domain of the function

To find the domain of the function \( g(t) = \sqrt{3 - t} - \sqrt{2 + t} \), we need to determine the values of \( t \) for which the function is defined. 

The function involves two square root terms: \( \sqrt{3 - t} \) and \( \sqrt{2 + t} \). For each square root to be defined, the expression inside the square root must be non-negative. Let's analyze each term separately:

1. **Condition for \( \sqrt{3 - t} \):**

   \[
   3 - t \geq 0
   \]

   Solving for \( t \):

   \[
   t \leq 3
   \]

2. **Condition for \( \sqrt{2 + t} \):**

   \[
   2 + t \geq 0
   \]

   Solving for \( t \):

   \[
   t \geq -2
   \]

Both conditions must be satisfied simultaneously. Therefore, we need to find the intersection of the intervals:

- \( t \leq 3 \)
- \( t \geq -2 \)

The intersection of these intervals is:

\[
-2 \leq t \leq 3
\]

Thus, the **domain of the function** \( g(t) \) is all \( t \) values in the closed interval \([-2, 3]\).

### Conclusion
The domain of \( g(t) = \sqrt{3 - t} - \sqrt{2 + t} \) is:

\[
\boxed{[-2, 3]}
\]

Would you like more details on the steps, or do you have any other questions?

Here are 5 related questions to expand your understanding:

1. What is the range of the function \( g(t) = \sqrt{3 - t} - \sqrt{2 + t} \) over its domain?
2. How would the domain change if the function were \( h(t) = \sqrt{3 - t} + \sqrt{2 + t} \)?
3. How can you graph the function \( g(t) = \sqrt{3 - t} - \sqrt{2 + t} \)?
4. What are the points where \( g(t) = 0 \) within the domain?
5. How would you find the derivative of \( g(t) = \sqrt{3 - t} - \sqrt{2 + t} \)?

**Tip:** When determining the domain of functions involving square roots, always set the expression inside the square root greater than or equal to zero to ensure real-valued outputs.

Discover how to find the domain of the function g(t) = sqrt(3 - t) - sqrt(2 + t). This problem involves determining the values of t where the function is defined, based on conditions for square root terms and inequalities. Suitable for high school students and those studying algebra and functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation