how to find the domain of this function"\sqrt{\left(x^{2}-3\right)+2}

To find the domain of the function \( f(x) = \sqrt{x^2 - 3 + 2} \), follow these steps:

1. **Simplify the expression inside the square root**:
   \[
   f(x) = \sqrt{x^2 - 3 + 2} = \sqrt{x^2 - 1}
   \]

2. **Determine the condition for the square root to be defined**:
   The square root function is only defined for non-negative values, so the expression inside the square root must be greater than or equal to zero:
   \[
   x^2 - 1 \geq 0
   \]

3. **Solve the inequality \( x^2 - 1 \geq 0 \)**:
   \[
   x^2 \geq 1
   \]
   This inequality holds when:
   \[
   x \leq -1 \quad \text{or} \quad x \geq 1
   \]

4. **Express the domain**:
   The domain of the function is the set of all \( x \) values that satisfy the inequality. In interval notation, this is:
   \[
   (-\infty, -1] \cup [1, \infty)
   \]

So, the domain of the function \( f(x) = \sqrt{x^2 - 1} \) is \( (-\infty, -1] \cup [1, \infty) \).

Would you like more details or have any questions? Here are five related questions you might explore:

1. What is the domain of \( \sqrt{x^2 + 4} \)?
2. How would the domain change if the function were \( \sqrt{-x^2 + 4} \)?
3. What is the range of the function \( f(x) = \sqrt{x^2 - 1} \)?
4. How do you find the domain of a rational function?
5. What happens to the domain if you add or subtract a constant inside the square root?

**Tip**: When finding the domain of functions involving square roots, always set the expression inside the square root to be greater than or equal to zero.

Learn how to find the domain of the function \( f(x) = \sqrt{x^2 - 3 + 2} \) step-by-step. Understand the conditions for the square root function to be defined and solve the corresponding inequality. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation