For the function $$ F(x)=\frac{x^{2}-6}{x^{2}} $$, (a) graph the rational function

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1. **Rewrite the Function**

We start with
   $$
   F(x) = \frac{x^2-6}{x^2}.
   $$
   This can be rewritten by splitting the fraction:
   $$
   F(x) = \frac{x^2}{x^2} - \frac{6}{x^2} = 1 - \frac{6}{x^2}.
   $$

2. **Determine the Domain**

The function is undefined when the denominator is zero, so set
   $$
   x^2 = 0 \quad \Longrightarrow \quad x = 0.
   $$
   Thus, the domain is all real numbers except $$x=0$$.

3. **Identify Intercepts**

- **$$x$$-intercepts:**

Set $$F(x)=0$$:
     $$
     1 - \frac{6}{x^2} = 0 \quad \Longrightarrow \quad \frac{6}{x^2} = 1 \quad \Longrightarrow \quad x^2 = 6.
     $$
     This gives
     $$
     x = \pm\sqrt{6}.
     $$
     
   - **$$y$$-intercept:**

A $$y$$-intercept would occur when $$x=0$$. However, $$x=0$$ is not in the domain. Therefore, there is no $$y$$-intercept.

4. **Determine Asymptotes**

- **Vertical Asymptote:**

A vertical asymptote occurs where the function is undefined and the limit tends to infinity (or negative infinity). Since $$x=0$$ is not in the domain, consider the limit:
     $$
     \lim_{x \to 0} F(x) = 1 - \frac{6}{x^2}.
     $$
     As $$x$$ approaches $$0$$ (from either side), $$\frac{6}{x^2}$$ tends to $$+\infty$$ and so
     $$
     F(x) \to -\infty.
     $$
     Hence, there is a vertical asymptote at $$x=0$$.

- **Horizontal Asymptote:**

Investigate the behavior as $$|x|$$ becomes large:
     $$
     \lim_{x \to \pm\infty} F(x) = 1 - \frac{6}{x^2} = 1.
     $$
     Thus, there is a horizontal asymptote at $$y=1$$.

5. **Analyze Symmetry**

Notice that $$F(x)$$ is an even function because
   $$
   F(-x) = 1 - \frac{6}{(-x)^2} = 1 - \frac{6}{x^2} = F(x).
   $$
   This means the graph is symmetric with respect to the $$y$$-axis.

6. **Plot Key Points**

Compute a few points to understand the graph:

- For $$x= \sqrt{6} \approx 2.45$$, we have $$F(\sqrt{6}) = 0$$.
   - For $$x=-\sqrt{6} \approx -2.45$$, we have $$F(-\sqrt{6}) = 0$$.
   - For $$x=1$$, 
     $$
     F(1) = 1 - \frac{6}{1^2} = 1 - 6 = -5.
     $$
   - For $$x=2$$,
     $$
     F(2) = 1 - \frac{6}{4} = 1 - 1.5 = -0.5.
     $$
   - For $$x=3$$,
     $$
     F(3) = 1 - \frac{6}{9} \approx 1 - 0.667 \approx 0.333.
     $$

7. **Graph Characteristics Summary**

- The function $$F(x) = 1 - \frac{6}{x^2}$$ is defined for all $$x \neq 0$$.
   - It has a vertical asymptote at $$x=0$$ where the function tends to $$-\infty$$ as $$x$$ approaches $$0$$.
   - There is a horizontal asymptote at $$y=1$$ as $$|x|$$ increases.
   - $$x$$-intercepts occur at $$x = \pm \sqrt{6}$$.
   - No $$y$$-intercept exists since the function is not defined at $$x=0$$.
   - The graph is symmetric with respect to the $$y$$-axis (even function).

8. **Sketch the Graph**

- Draw a dashed vertical line at $$x=0$$ to represent the vertical asymptote.
   - Draw a dashed horizontal line at $$y=1$$ to represent the horizontal asymptote.
   - Plot the points $$(\sqrt{6}, 0)$$ and $$(-\sqrt{6}, 0)$$.
   - Plot the key points such as $$(1, -5)$$, $$(2, -0.5)$$, and $$(3, 0.333)$$, remembering to mirror them across the $$y$$-axis.
   - Notice the curve approaches the asymptotes: as $$x$$ approaches $$0$$, the graph drops steeply downward, and as $$|x|$$ becomes large, the graph levels off near $$y=1$$.

This step-by-step analysis provides all information needed to sketch the graph of the function $$ F(x)=\frac{x^{2}-6}{x^{2}} $$.
To graph the function $$ F(x) = \frac{x^2 - 6}{x^2} $$: 1. **Rewrite the Function:** $$ F(x) = 1 - \frac{6}{x^2} $$ 2. **Domain:** - All real numbers except $$ x = 0 $$. 3. **Intercepts:** - $$ x $$-intercepts at $$ x = \sqrt{6} $$ and $$ x = -\sqrt{6} $$. - No $$ y $$-intercept. 4. **Asymptotes:** - Vertical asymptote at $$ x = 0 $$. - Horizontal asymptote at $$ y = 1 $$. 5. **Symmetry:** - The function is even, symmetric about the $$ y $$-axis. 6. **Key Points:** - $$ F(1) = -5 $$ - $$ F(2) = -0.5 $$ - $$ F(3) \approx 0.333 $$ 7. **Graph Characteristics:** - Approaches the vertical asymptote at $$ x = 0 $$ from both sides. - Approaches the horizontal asymptote $$ y = 1 $$ as $$ |x| $$ increases. - Crosses the $$ x $$-axis at $$ \sqrt{6} $$ and $$ -\sqrt{6} $$. 8. **Sketch the Graph:** - Draw vertical and horizontal asymptotes. - Plot intercepts and key points. - Mirror points across the $$ y $$-axis due to symmetry. - Ensure the graph levels off near $$ y = 1 $$ for large $$ |x| $$ and approaches $$ y = -\infty $$ near $$ x = 0 $$. This summary provides all necessary information to accurately sketch the graph of the given function.

For the function \( F(x)=\frac{x^{2}-6}{x^{2}} \), (a) graph the rational function

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