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Understanding the behavior of a rational function is the first step in graphing it accurately. The function given here is \( y = \frac{1}{x^2} \). For this type of function, you'll notice some distinct characteristics:

• As \( x \) gets very close to 0 from either side, \( y \) becomes very large. This is what we call a "vertical asymptote" at \( x = 0 \).
• The function is always positive, meaning all outputs (\( y \)-values) are greater than zero, regardless of whether \( x \) is positive or negative.

These observations help us predict that the graph will rise steeply as it approaches \( x = 0 \) from both the negative and positive directions. The graph does not touch or cross the \( x \)-axis due to the absence of any real \( x \) value that could make the function zero.

Finding the domain and range of a function provides essential boundaries for graphing. For \( y = \frac{1}{x^2} \), the domain is all real numbers except \( x = 0 \), because division by zero is undefined. In practical terms, for this exercise, we're graphing the function in intervals \([-0.5, -0.1] \cup [0.1, 0.5]\).
The range is even more straightforward. Since \( y \) is the result of dividing 1 by a positive number (\( x^2 \)), the smallest \( y \) can be is 4 when \( x = \pm 0.5 \). Conversely, the largest \( y \) value occurs when \( x \) is closest to zero; at \( x = \pm 0.1 \), \( y \) becomes 100. Thus, the range over this domain is \([4, 100]\).

• Domain: \( x \in (-\infty, 0) \cup (0, \infty) \) but focused on \([-0.5, -0.1]\) and \([0.1, 0.5]\).
• Range: \([4, 100]\)

The range gives useful information about how high or low the graph will go within those domain segments.

Plotting points is a practical approach to sketch the graph of a function. For \( y = \frac{1}{x^2} \), start by calculating specific points:

• At \( x = -0.5 \), \( y = 4 \)
• At \( x = -0.1 \), \( y = 100 \)
• At \( x = 0.1 \), \( y = 100 \)
• At \( x = 0.5 \), \( y = 4 \)

These points indicate where the graph will be at specific \( x \)-values within our domain. The steep rise as \( x \) moves towards zero accounts for the high \( y \)-values near \( x = \pm 0.1 \). By connecting these points within their respective intervals \([-0.5, -0.1]\) and \([0.1, 0.5]\), you map the graph accurately.
Remember:

• Use clear intervals and consistent scaling.
• Highlight both sides of the function, indicating the steep increase towards the asymptote at \( x = 0 \).

Effective plotting provides clarity and validates the behavior and limits of the function you calculated initially.