For the function $$ F(x)=\frac{x^{2}-5}{x^{2}} $$, (a) graph the rational function

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1. **Rewrite the function** The function is given by $$ F(x)=\frac{x^{2}-5}{x^{2}}. $$ This can be rewritten as $$ F(x)=\frac{x^{2}}{x^{2}}-\frac{5}{x^{2}}=1-\frac{5}{x^{2}}. $$ 2. **Determine the domain** The denominator $$x^2$$ cannot be zero. Hence, $$ x^2 \ne 0 \quad \Longrightarrow \quad x \ne 0. $$ So, the domain is $$\{x\in\mathbb{R} : x\ne 0\}$$. 3. **Identify intercepts** - **$$x$$-intercepts:** Set $$F(x)=0$$: $$ 1-\frac{5}{x^{2}}=0 \quad \Longrightarrow \quad \frac{5}{x^{2}}=1 \quad \Longrightarrow \quad x^{2}=5. $$ This gives $$ x=\sqrt{5} \quad \text{and} \quad x=-\sqrt{5}. $$ - **$$y$$-intercept:** The $$y$$-intercept would occur at $$x=0$$. However, since $$x=0$$ is not in the domain, there is no $$y$$-intercept. 4. **Find asymptotes** - **Vertical asymptote:** Since the function is undefined at $$x=0$$ and the values of $$F(x)$$ approach $$-\infty$$ as $$x \to 0$$ (see behavior below), there is a vertical asymptote at $$ x=0. $$ - **Horizontal asymptote:** As $$x$$ becomes very large in absolute value, the term $$\frac{5}{x^2}$$ tends to $$0$$: $$ \lim_{x\to \pm\infty} F(x)=1-\lim_{x\to \pm\infty}\frac{5}{x^2}=1. $$ Hence, the horizontal asymptote is $$ y=1. $$ 5. **Analyze symmetry** Replace $$x$$ with $$-x$$ in $$F(x)$$: $$ F(-x)=1-\frac{5}{(-x)^2}=1-\frac{5}{x^2}=F(x). $$ Thus, the function is even and symmetric about the $$y$$-axis. 6. **Behavior near the asymptotes and intercepts** - **Near the vertical asymptote $$x=0$$:** As $$x\to 0^+$$ or $$x\to 0^-$$, the term $$\frac{5}{x^2}$$ becomes very large (positive), so $$ F(x)=1-\frac{5}{x^2}\to -\infty. $$ - **Near the horizontal asymptote $$y=1$$:** As $$x\to \pm\infty$$, $$\frac{5}{x^2}$$ becomes negligible, and $$F(x)$$ approaches $$1$$ from below because $$\frac{5}{x^2}$$ is always positive. 7. **Sketching the graph** - **For $$x>0$$:** - The graph has a vertical asymptote at $$x=0$$ where $$F(x)\to -\infty$$. - It crosses the $$x$$-axis at $$x=\sqrt{5}$$. - As $$x\to+\infty$$, it approaches the horizontal asymptote $$y=1$$ from below. - **For $$x<0$$:** - Similarly, there is a vertical asymptote at $$x=0$$ with $$F(x)\to -\infty$$. - The graph crosses the $$x$$-axis at $$x=-\sqrt{5}$$. - As $$x\to-\infty$$, it approaches the horizontal asymptote $$y=1$$ from below. - **Symmetry:** Because the function is even, the left-hand graph is a mirror image of the right-hand graph. 8. **Graph description** The graph consists of two branches: - One branch in the first quadrant (for $$x>0$$) starts very low (tending to $$-\infty$$) as $$x$$ approaches $$0$$, increases, crosses the $$x$$-axis at $$x=\sqrt{5}$$, and then levels off toward the horizontal line $$y=1$$. - The other branch in the second quadrant (for $$x<0$$) behaves similarly: it goes down to $$-\infty$$ near $$x=0$$, crosses the $$x$$-axis at $$x=-\sqrt{5}$$, and approaches $$y=1$$ as $$x$$ decreases (i.e., as $$|x|$$ increases). The graph in summary: - Domain: $$ x \in \mathbb{R}\setminus\{0\} $$ - Vertical asymptote: $$ x=0 $$ - Horizontal asymptote: $$ y=1 $$ - $$x$$-intercepts: $$ x=\pm\sqrt{5} $$ - No $$y$$-intercept. - Even function, symmetric about the $$y$$-axis. Using this information, one can plot the branches accurately on the coordinate plane. The function $$ F(x) = \frac{x^2 - 5}{x^2} $$ can be graphed as follows: - **Domain:** All real numbers except $$ x = 0 $$. - **Intercepts:** $$ x $$-intercepts at $$ x = \sqrt{5} $$ and $$ x = -\sqrt{5} $$; no $$ y $$-intercept. - **Asymptotes:** Vertical asymptote at $$ x = 0 $$ and horizontal asymptote at $$ y = 1 $$. - **Symmetry:** The function is even, symmetric about the $$ y $$-axis. - **Behavior:** As $$ x $$ approaches 0 from either side, $$ F(x) $$ tends to $$-\infty$$. As $$ x $$ approaches $$ \pm\infty $$, $$ F(x) $$ approaches 1. The graph consists of two branches symmetric about the $$ y $$-axis, crossing the $$ x $$-axis at $$ \sqrt{5} $$ and $$ -\sqrt{5} $$, and approaching the horizontal asymptote $$ y = 1 $$ as $$ x $$ moves away from 0.

For the function \( F(x)=\frac{x^{2}-5}{x^{2}} \), (a) graph the rational function

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