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.