For the function $$ F(x)=\frac{x^{2}-3}{x^{2}} $$, (a) graph the rational function

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**Step 1. Rewrite the Function** We start with $$ F(x)=\frac{x^2-3}{x^2}. $$ This can be rewritten as $$ F(x)=\frac{x^2}{x^2}-\frac{3}{x^2}=1-\frac{3}{x^2}. $$ **Step 2. Determine the Domain** The function is defined wherever the denominator is not zero. Since $$x^2=0$$ when $$x=0$$, the domain is $$ \{x \in \mathbb{R} : x\neq 0\}. $$ **Step 3. Identify Asymptotes** - **Vertical Asymptote:** The denominator $$x^2$$ is zero at $$x=0$$; hence, there is a vertical asymptote at $$ x=0. $$ - **Horizontal Asymptote:** For large values of $$|x|$$, the term $$\frac{3}{x^2}$$ approaches $$0$$. Thus, $$ \lim_{x\to\pm\infty} F(x)=1. $$ Therefore, the horizontal asymptote is $$ y=1. $$ **Step 4. Find Intercepts** - **$$x$$-Intercepts:** The $$x$$-intercepts occur when $$F(x)=0$$: $$ 1-\frac{3}{x^2}=0 \quad \Longrightarrow \quad \frac{3}{x^2}=1 \quad \Longrightarrow \quad x^2=3. $$ Therefore, $$ x=\sqrt{3} \quad \text{or} \quad x=-\sqrt{3}. $$ - **$$y$$-Intercept:** To find the $$y$$-intercept, set $$x=0$$. However, $$x=0$$ is not in the domain (vertical asymptote), so there is no $$y$$-intercept. **Step 5. Check for Symmetry** Since the function can be rewritten as $$ F(x)=1-\frac{3}{x^2}, $$ and the term $$\frac{3}{x^2}$$ is an even function (i.e., it satisfies $$ \frac{3}{(-x)^2}=\frac{3}{x^2}$$), we have $$ F(-x)=F(x). $$ Thus, the function is even and symmetric about the $$y$$-axis. **Step 6. Analyze Behavior Near the Vertical Asymptote** Examine the limits as $$x$$ approaches 0: - As $$x\to 0^+$$ (from the right), $$ \frac{3}{x^2}\to +\infty \quad \text{so} \quad F(x)=1-\frac{3}{x^2}\to -\infty. $$ - As $$x\to 0^-$$ (from the left), $$ \frac{3}{x^2}\to +\infty \quad \text{so} \quad F(x)=1-\frac{3}{x^2}\to -\infty. $$ Thus, on both sides of $$x=0$$, the function decreases without bound. **Step 7. Sketching the Graph** - **For $$x>0$$:** - The graph has a vertical asymptote at $$x=0$$ and a horizontal asymptote at $$y=1$$. - It crosses the $$x$$-axis at $$x=\sqrt{3}$$. - As $$x\to 0^+$$, $$F(x)\to -\infty$$. - As $$x\to +\infty$$, $$F(x)\to 1$$ from below. - **For $$x<0$$:** - By symmetry, the graph is a mirror image of the right side. - There is a vertical asymptote at $$x=0$$ and a horizontal asymptote at $$y=1$$. - The graph crosses the $$x$$-axis at $$x=-\sqrt{3}$$. - As $$x\to 0^-$$, $$F(x)\to -\infty$$. - As $$x\to -\infty$$, $$F(x)\to 1$$ from below. **Step 8. Graph Features Summary** - **Domain:** $$x\neq 0$$ - **Vertical Asymptote:** $$x=0$$ - **Horizontal Asymptote:** $$y=1$$ - **$$x$$-Intercepts:** $$x=-\sqrt{3}$$ and $$x=\sqrt{3}$$ - **No $$y$$-Intercept** - **Even Function (symmetric about the $$y$$-axis)** - **Behavior Near $$x=0$$:** $$F(x)\to -\infty$$ from both sides **Final Graph Description** - Draw a dashed vertical line at $$x=0$$ to represent the vertical asymptote. - Draw a dashed horizontal line at $$y=1$$ for the horizontal asymptote. - Plot the points $$(-\sqrt{3},0)$$ and $$(\sqrt{3},0)$$ where the graph crosses the $$x$$-axis. - The branches for $$x>0$$ and $$x<0$$ both approach $$y=1$$ as $$|x|$$ increases, coming from below (since $$F(x) < 1$$ for all $$x\neq 0$$). - As $$x$$ approaches 0 from either side, the function goes downward to negative infinity. These steps fully detail how to graph the function $$ F(x)=\frac{x^2-3}{x^2} $$. To graph the function $$ F(x) = \frac{x^2 - 3}{x^2} $$: 1. **Rewrite the Function:** $$ F(x) = 1 - \frac{3}{x^2} $$ 2. **Determine the Domain:** - $$ x \neq 0 $$ 3. **Identify Asymptotes:** - **Vertical Asymptote:** $$ x = 0 $$ - **Horizontal Asymptote:** $$ y = 1 $$ 4. **Find Intercepts:** - **$$ x $$-Intercepts:** $$ x = \sqrt{3} $$ and $$ x = -\sqrt{3} $$ - **No $$ y $$-Intercept** 5. **Symmetry:** - The function is even, symmetric about the $$ y $$-axis. 6. **Behavior Near Asymptote:** - As $$ x $$ approaches 0 from either side, $$ F(x) $$ approaches negative infinity. 7. **Graph Summary:** - Vertical asymptote at $$ x = 0 $$ - Horizontal asymptote at $$ y = 1 $$ - Crosses the $$ x $$-axis at $$ x = \sqrt{3} $$ and $$ x = -\sqrt{3} $$ - Approaches $$ y = 1 $$ as $$ |x| $$ increases - Symmetric about the $$ y $$-axis **Final Graph:** - Draw vertical and horizontal asymptotes. - Plot $$ x $$-intercepts at $$ \sqrt{3} $$ and $$ -\sqrt{3} $$. - Sketch the branches approaching $$ y = 1 $$ from below and approaching negative infinity near $$ x = 0 $$.

For the function \( F(x)=\frac{x^{2}-3}{x^{2}} \), (a) graph the rational function

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