Let's analyze the function $f(x) = \frac{3x - 2}{x^2}$ to determine its domain and range.

Domain:

The domain of a function consists of all the values of $x$ for which the function is defined.

For $f(x) = \frac{3x - 2}{x^2}$, the function is undefined when the denominator is zero. The denominator here is $x^2$, and $x^2 = 0$ when $x = 0$.

Thus, the function is undefined at $x = 0$.

Domain: $x \in \mathbb{R} \setminus \{0\}$, or in interval notation: $(-\infty, 0) \cup (0, \infty)$.

Range:

To find the range, we analyze the behavior of the function as $x$ varies over its domain.

1. For large positive or negative values of $x$, $f(x)$ behaves approximately like $\frac{3x}{x^2} = \frac{3}{x}$, which approaches 0 as $x$ tends to $\pm\infty$.

2. Near $x = 0$, the function's behavior depends on whether $x$ approaches 0 from the positive side or the negative side:

   • As $x \to 0^+$ (from the right), $x^2$ is small and positive, making $f(x)$ large and positive.
   • As $x \to 0^-$ (from the left), $x^2$ is still small and positive, but the numerator $3x - 2$ is negative, so $f(x)$ becomes large and negative.

This suggests that the function can take any real value as $x$ approaches 0 from either side and as $x$ tends to $\pm\infty$.

Range: $f(x) \in \mathbb{R}$, or in interval notation: $(-\infty, \infty)$.

Summary:

• Domain: $(-\infty, 0) \cup (0, \infty)$
• Range: $(-\infty, \infty)$

Would you like more details or have any questions?

Here are some related questions:

1. What happens if the numerator of a rational function is zero?
2. How does the behavior of a function near a vertical asymptote affect its range?
3. How can the domain of a function change if the expression is more complex, like a square root?
4. Can a function have a restricted range even if its domain is all real numbers?
5. What is the impact of adding or subtracting constants on the domain and range of a function?

Tip: Always check for undefined points in a function's domain, especially where the denominator equals zero or inside square roots and logarithms.