Find the domain and range of 1/root(25-x²)

To find the **domain** and **range** of the function \( f(x) = \frac{1}{\sqrt{25 - x^2}} \), we need to ensure that the expression inside the square root is non-negative, and that the denominator is non-zero.

### 1. **Domain**:
The function is defined when:
- The expression inside the square root, \( 25 - x^2 \), is non-negative, meaning \( 25 - x^2 \geq 0 \).
- The denominator must not be zero, so \( \sqrt{25 - x^2} \neq 0 \).

Let's solve these conditions step by step.

#### Step 1: Solve \( 25 - x^2 \geq 0 \):
\[
25 \geq x^2 \implies -5 \leq x \leq 5
\]
This gives the interval \( x \in [-5, 5] \), which ensures the square root is non-negative.

#### Step 2: Ensure the denominator is non-zero:
For the denominator to be non-zero, \( \sqrt{25 - x^2} \neq 0 \), which means \( 25 - x^2 \neq 0 \). This simplifies to:
\[
x^2 \neq 25 \implies x \neq \pm 5
\]

Thus, the domain is all \( x \) such that \( -5 < x < 5 \).

#### Final Domain:
\[
x \in (-5, 5)
\]

### 2. **Range**:
To find the range of \( f(x) = \frac{1}{\sqrt{25 - x^2}} \), we need to consider the behavior of the function within its domain.

- As \( x \) approaches \( -5 \) or \( 5 \), the term \( \sqrt{25 - x^2} \) approaches zero, meaning \( f(x) \) approaches infinity.
- As \( x \) approaches 0, \( \sqrt{25 - 0^2} = \sqrt{25} = 5 \), so \( f(0) = \frac{1}{5} \).

Thus, the function can take any value from \( \frac{1}{5} \) to \( +\infty \).

#### Final Range:
\[
f(x) \in \left( \frac{1}{5}, +\infty \right)
\]

### Summary:
- **Domain**: \( (-5, 5) \)
- **Range**: \( \left( \frac{1}{5}, +\infty \right) \)

Would you like more details or have any questions? Here are some related questions:

1. How do we handle square roots in other types of functions?
2. What happens when \( x \) is exactly 5 or -5?
3. How do we find the range of a rational function in general?
4. What transformations would shift the domain and range of this function?
5. How would the domain and range change if the square root expression were \( 25 - x^4 \)?

**Tip**: Always check both the non-negativity of square roots and the non-zero condition for denominators when dealing with such functions.

Find the domain and range of 1/root(25 - x²)

Discover how to find the domain and range of the function 1/root(25 - x²). This problem involves understanding the conditions for a square root and rational function to be defined, and finding the appropriate domain and range for students in Grades 10-12.

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