For the function \( F(x)=\frac{x^{2}-6}{x^{2}} \), (a) graph the rational function
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To graph the rational function \( F(x) = \frac{x^2 - 6}{x^2} \), start by identifying key features. The function can be simplified to \( F(x) = 1 - \frac{6}{x^2} \). This indicates that as \( x \) approaches 0, \( F(x) \) heads towards negative infinity, indicating a vertical asymptote at \( x = 0 \). Additionally, the horizontal asymptote is \( y = 1 \), as \( x \) approaches ±∞. So, plot points around these features to create a smooth curve that approaches the asymptotes. Next, look for intercepts: for the x-intercept, set \( F(x) = 0 \) which gives \( x^2 - 6 = 0 \) leading to \( x = ±\sqrt{6} \), and for the y-intercept, evaluate \( F(0) \), which confirms the vertical asymptote. Plotting these points and asymptotes provides a clear picture of the function’s behavior across the coordinate plane!