state the domain of each function h(x)=sqrt(6-x^2)

To determine the domain of the function \( h(x) = \sqrt{6 - x^2} \), we need to consider the conditions under which the expression inside the square root is non-negative, because the square root of a negative number is not a real number.

The expression inside the square root is \( 6 - x^2 \). Therefore, we require:

\[
6 - x^2 \geq 0
\]

This inequality can be rewritten as:

\[
x^2 \leq 6
\]

Taking the square root of both sides:

\[
-\sqrt{6} \leq x \leq \sqrt{6}
\]

So, the domain of the function \( h(x) = \sqrt{6 - x^2} \) is:

\[
x \in [-\sqrt{6}, \sqrt{6}]
\]

This means that \( h(x) \) is defined for all \( x \) values in the interval \([- \sqrt{6}, \sqrt{6}]\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the domain change if the function was \( h(x) = \sqrt{9 - x^2} \)?
2. What is the range of the function \( h(x) = \sqrt{6 - x^2} \)?
3. How do you find the domain of a function that includes a fraction?
4. What would be the domain if the function were \( h(x) = \sqrt{x^2 - 6} \) instead?
5. Can the function \( h(x) = \sqrt{6 - x^2} \) have negative values?

**Tip:** When finding the domain of a function involving a square root, always ensure the expression inside the square root is greater than or equal to zero.

Explore how to determine the domain of the function h(x) = sqrt(6 - x^2). Learn the conditions under which the square root function is defined, involving inequalities and square root properties. Suitable for Grades 10-12 students and anyone interested in understanding function domains.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation