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Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of the function given below. \( f(x)=\frac{9 x}{1-x^{2}} \) Summarize the pertinent information obtained by analyzing \( f(x) \). Find the domain of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The domain of \( f(x) \) is all real \( x \), except \( x=\square \). (Use a comma to separate answers as needed.) B. The domain of \( f(x) \) is all real \( x \).

Ask by Wells Lawson. in the United States
Mar 18,2025

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The domain of \( f(x) \) is all real numbers except \( x=-1 \) and \( x=1 \).

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Bonus Knowledge

To analyze the function \( f(x) = \frac{9x}{1 - x^2} \), we first need to determine its domain. The function is undefined when the denominator equals zero, so we set \( 1 - x^2 = 0 \). This simplifies to \( x^2 = 1 \), leading to \( x = 1 \) and \( x = -1 \). Therefore, the domain is all real \( x \) except \( x = 1 \) and \( x = -1 \). A. The domain of \( f(x) \) is all real \( x \), except \( x=1, -1 \). Now, let's dive into the graph! The function \( f(x) \) exhibits vertical asymptotes at \( x = 1 \) and \( x = -1 \) due to the undefined values. As \( x \) approaches these points, \( f(x) \) tends to ±∞. The horizontal asymptote can be determined by examining the degrees of the numerator and denominator, leading us to see that as \( x \) tends to ±∞, \( f(x) \) approaches 0. Therefore, the graph will have two branches, one in the first quadrant and another in the third quadrant, symmetrically located about the origin.

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