To find the domain and range of the function $f(x) = \frac{x^2}{1 - x^2}$, we need to determine the values of $x$ for which the function is defined and the set of possible values that the function can take.

Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

For $f(x) = \frac{x^2}{1 - x^2}$, the function is undefined when the denominator is zero. Therefore, we need to find the values of $x$ that make the denominator zero:

$1 - x^2 = 0$

Solving for $x$:

$x^2 = 1$ $x = \pm 1$

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

$\text{Domain} = \{ x \in \mathbb{R} \mid x \neq 1, x \neq -1 \}$

Range

The range of a function is the set of all possible output values (y-values) that the function can take.

To find the range of $f(x)$, we analyze the behavior of the function as $x$ approaches certain critical points:

1. As $x \to \pm \infty$: $\lim_{x \to \pm \infty} \frac{x^2}{1 - x^2} = \lim_{x \to \pm \infty} \frac{1}{\frac{1}{x^2} - 1} = \lim_{x \to \pm \infty} \frac{1}{0 - 1} = -1$ So, as $x$ approaches $\pm \infty$, $f(x)$ approaches -1.

2. At the vertical asymptotes $x = 1$ and $x = -1$: $\lim_{x \to 1^-} \frac{x^2}{1 - x^2} = \infty \quad \text{and} \quad \lim_{x \to 1^+} \frac{x^2}{1 - x^2} = -\infty$ $\lim_{x \to -1^-} \frac{x^2}{1 - x^2} = \infty \quad \text{and} \quad \lim_{x \to -1^+} \frac{x^2}{1 - x^2} = -\infty$ So, near the vertical asymptotes $x = 1$ and $x = -1$, the function takes on all values from $-\infty$ to $\infty$.

3. Behavior between and outside the critical points $x = 1$ and $x = -1$: For $-1 < x < 1$: $\frac{x^2}{1 - x^2} \text{ is positive and increases from 0 to } \infty$ For $x < -1$ or $x > 1$: $\frac{x^2}{1 - x^2} \text{ is negative and decreases from } -1 \text{ to } -\infty$

Combining these observations, we see that the range of $f(x)$ covers all real numbers except for $-1$:

$\text{Range} = \{ y \in \mathbb{R} \mid y \neq -1 \}$

Summary

• Domain: $\{ x \in \mathbb{R} \mid x \neq 1, x \neq -1 \}$
• Range: $\{ y \in \mathbb{R} \mid y \neq -1 \}$

Would you like further details or have any questions?

Here are five related questions you might find useful:

1. How do you determine the vertical asymptotes of a function?
2. How do you find the horizontal asymptotes of a rational function?
3. What are the steps to find the critical points of a function?
4. How do you determine the intervals of increase and decrease for a function?
5. What is the importance of analyzing the behavior of a function at infinity?

Tip: When finding the range of a function, it's often helpful to consider the function's behavior as $x$ approaches critical values and infinity.