find the domain and range of a f(x)=3x-2/x²

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.

Explore the domain and range of the rational function f(x) = (3x - 2) / x^2. Understand why the function is undefined at x = 0 and how the function behaves near asymptotic points. Suitable for high school students learning about rational functions and function analysis.

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