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

Ask by Bartlett Wyatt. in the United States
Mar 16,2025

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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.

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Bonus Knowledge

To graph the function \( F(x) = \frac{x^{2} - 5}{x^{2}} \), start by identifying its key characteristics, such as intercepts and asymptotes. The function has vertical asymptotes where the denominator is zero (at \( x = 0 \)), and it can be simplified to \( F(x) = 1 - \frac{5}{x^{2}} \). The horizontal asymptote is \( y = 1 \) since, as \( x \) approaches infinity or negative infinity, the term \( \frac{5}{x^{2}} \) approaches zero. Your graph will show the function approaching the horizontal line at \( y = 1 \) and diverging as it nears \( x = 0 \). For plotting, calculate key points such as \( F(1) = -4 \), \( F(2) = -1.25 \), and \( F(-1) = -4 \). Marking these points will help illustrate the decrease to the vertical asymptote and overall shape of the graph. Don’t forget to consider the symmetry about the y-axis, as the function is even! With key points and asymptotic behavior in mind, sketch a smooth curve that adheres to the identified characteristics.

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