what is the domain of the function g(x) = root over 1-x^2


The function \( g(x) = \sqrt{1 - x^2} \) involves a square root, which means we need to ensure that the expression inside the square root is non-negative. This constraint is essential because the square root of a negative number is not defined in the real number system.

To find the domain, let's consider the condition:

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

### Step 1: Solve the inequality

1. Rearrange the inequality to:
   \[
   x^2 \leq 1
   \]

2. Take the square root of both sides:
   \[
   -1 \leq x \leq 1
   \]

### Step 2: Determine the domain

The domain of \( g(x) \) is all values of \( x \) that satisfy \( -1 \leq x \leq 1 \). This ensures that \( 1 - x^2 \) remains non-negative, allowing the square root to be defined.

### Conclusion

The domain of \( g(x) = \sqrt{1 - x^2} \) is:

\[
[-1, 1]
\]

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. What is the range of \( g(x) = \sqrt{1 - x^2} \)?
2. How would the domain change if the function were \( h(x) = \sqrt{x^2 - 1} \)?
3. How can we find the domain of functions involving higher roots, like \( f(x) = \sqrt[3]{1 - x^2} \)?
4. What would the graph of \( g(x) = \sqrt{1 - x^2} \) look like on a coordinate plane?
5. How do we find the domain of a function involving both square roots and fractions?

**Tip:** For functions with square roots, always ensure the expression under the square root is non-negative to determine the domain.

What is the domain of the function g(x) = root over 1 - x^2?

Discover how to determine the domain of the function g(x) = √(1 - x^2), including step-by-step guidance on solving inequalities to ensure the square root expression remains non-negative. Essential algebra concepts covered for students in Grades 9-11.

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