To find the domain and range of the function \( f(x) = \frac{x^2}{1 - x^2} \), we need to consider where the function is defined (domain) and the set of possible output values (range).

### Domain

The domain of the function is the set of all \( x \) values for which the function is defined. The function \( f(x) = \frac{x^2}{1 - x^2} \) has a denominator \( 1 - x^2 \). The denominator cannot be zero because division by zero is undefined. Therefore, we need to find where \( 1 - x^2 = 0 \):

\[ 1 - x^2 = 0 \]
\[ x^2 = 1 \]
\[ x = \pm 1 \]

Thus, the function is undefined at \( x = 1 \) and \( x = -1 \). Therefore, the domain is all real numbers except \( x = 1 \) and \( x = -1 \):

\[ \text{Domain: } x \in (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \]

### Range

To find the range, we analyze the behavior of the function:

1. As \( x \) approaches \( 1 \) from the left, \( 1 - x^2 \) approaches \( 0 \) from the positive side, making \( f(x) \) approach \( +\infty \).
2. As \( x \) approaches \( 1 \) from the right, \( 1 - x^2 \) approaches \( 0 \) from the negative side, making \( f(x) \) approach \( -\infty \).
3. As \( x \) approaches \( -1 \) from the left, \( 1 - x^2 \) approaches \( 0 \) from the positive side, making \( f(x) \) approach \( +\infty \).
4. As \( x \) approaches \( -1 \) from the right, \( 1 - x^2 \) approaches \( 0 \) from the negative side, making \( f(x) \) approach \( -\infty \).

We observe that the function can take any real value except it does not have any finite limits at \( x = 1 \) and \( x = -1 \). 

Therefore, the range of the function is all real numbers:

\[ \text{Range: } (-\infty, \infty) \]

### Summary

- **Domain:** \( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \)
- **Range:** \( (-\infty, \infty) \)

Would you like more details or have any questions?

Here are 5 numbered questions for you to ask:

1. How do you find the domain of a function involving a square root?
2. What is the range of the function \( f(x) = \frac{1}{x} \)?
3. How do you find the asymptotes of a rational function?
4. Can you explain how to determine the inverse of a function?
5. What is the domain of the function \( g(x) = \sqrt{x - 2} \)?

**Tip:** When finding the domain of a function, always look for values that make the denominator zero or the argument of a square root negative.

Learn how to find the domain and range of the function f(x) = x² / (1 - x²) with a detailed explanation. Understand the conditions for the function's domain and the complete set of possible output values. Suitable for advanced high school students.

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