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

### 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 \( 0 \), \( f(x) \) approaches \( 0 \).
2. As \( x \) approaches \( 1 \) from the left, \( f(x) \) approaches \( +\infty \).
3. As \( x \) approaches \( 1 \) from the right, \( f(x) \) approaches \( -\infty \).
4. As \( x \) approaches \( -1 \) from the left, \( f(x) \) approaches \( +\infty \).
5. As \( x \) approaches \( -1 \) from the right, \( f(x) \) approaches \( -\infty \).
6. For large positive or negative values of \( x \), \( f(x) \) approaches \( -1 \).

Next, we solve for \( y \) in terms of \( x \):

\[ y = \frac{x^2}{1 - x^2} \]
\[ y(1 - x^2) = x^2 \]
\[ y - yx^2 = x^2 \]
\[ y = x^2(1 + y) \]
\[ x^2 = \frac{y}{1 + y} \]

For \( x^2 \) to be non-negative, \( \frac{y}{1 + y} \geq 0 \). This implies \( y \geq 0 \) or \( y \leq -1 \).

Therefore, the range of the function is:

\[ \text{Range: } (-\infty, -1] \cup [0, \infty) \]

### Summary

- **Domain:** \( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \)
- **Range:** \( (-\infty, -1] \cup [0, \infty) \)

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.

Explore how to determine the domain and range of the rational function f(x) = x² / (1 - x²). This detailed explanation covers the function's domain restrictions and the set of possible output values. Suitable for advanced high school students studying functions and rational expressions.

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