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The domain of a function represents all the possible values of the variable for which the function is defined. For the function \(f(x) = \frac{1}{x^2}\), the domain excludes \(x = 0\), where the function is undefined due to division by zero. Therefore, the domain is \((-\infty, 0) \cup (0, \infty)\).

The range of a function is the set of all possible output values. For \(f(x) = \frac{1}{x^2}\), the output is always positive since square of any real number is non-negative, making the range \((0, \infty)\).

Understanding domain and range is crucial in comprehending the limitations and behavior of any given function.

• Domain: All possible \(x\)-values. For this function: \((-\infty, 0) \cup (0, \infty)\)
• Range: All possible \(f(x)\)-values. For this function: \((0, \infty)\)

Asymptotes are lines that the graph of a function approaches but never actually meets. They provide significant insights into the behavior of the graph at extreme values. For \(f(x) = \frac{1}{x^2}\), there are two types of asymptotes to identify:

- **Vertical Asymptote:** This occurs at \(x = 0\) because the function becomes undefined as \(x\) approaches zero. As \(x\to 0^+\) or \(x\to 0^-\), \(f(x)\) tends towards infinity. Thus, there's a vertical asymptote along this line.

- **Horizontal Asymptote:** At extreme values of \(x\) (as \(x\to\pm \infty\)), the function \(f(x)\) tends towards zero, but never actually reaches zero, resulting in a horizontal asymptote at \(y = 0\).

Recognizing asymptotes informs how the graph stretches towards infinity and how the function behaves at its boundary values.

• Vertical Asymptote: \(x = 0\)
• Horizontal Asymptote: \(y = 0\)

When analyzing a function, it's useful to determine where it's increasing or decreasing.

For \(f(x) = \frac{1}{x^2}\), as \(x\) moves from \(-\infty\) to 0, and from 0 to \(+\infty\), the function is decreasing. This means that as we move along the x-values, the output of the function gets smaller.

Remember:

• A function is increasing if the values of \(f(x)\) rise as \(x\) increases.
• A function is decreasing if the values of \(f(x)\) drop as \(x\) rises.

For this function:
- **Decreasing interval:** Both \((-\infty, 0)\) and \((0, \infty)\). The function decreases in both directions away from zero.

Understanding these intervals helps in predicting the trend of the function as \(x\) varies.