1. The problem is to understand and analyze the function $f(x) = \frac{1}{x^2}$. 2. This function is a rational function where the numerator is 1 and the denominator is $x^2$. 3. Important rules: The function is undefined at $x=0$ because division by zero is undefined. The function is always positive since $x^2$ is always positive except at zero. 4. To analyze the behavior, consider the limits: $$\lim_{x \to 0^+} \frac{1}{x^2} = +\infty$$ $$\lim_{x \to 0^-} \frac{1}{x^2} = +\infty$$ 5. As $x$ approaches zero from either side, the function tends to infinity, indicating a vertical asymptote at $x=0$. 6. For large values of $x$, $$\lim_{x \to \pm \infty} \frac{1}{x^2} = 0$$ 7. This means the function approaches zero as $x$ goes to positive or negative infinity, indicating a horizontal asymptote at $y=0$. 8. The function is symmetric about the y-axis because $f(-x) = f(x)$, so it is an even function. 9. The graph has a minimum value at $x=\pm \infty$ approaching zero but no maximum value since it goes to infinity near zero. 10. Summary: The function $f(x) = \frac{1}{x^2}$ has a vertical asymptote at $x=0$, a horizontal asymptote at $y=0$, is always positive, and is even symmetric.