ummarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( f(x)=\frac{x+3}{x-3} \) ind any horizontal asymptotes of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, \( \square \) (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is \( \square \) and the bottom asymptote is \( \square \) (Type equations.) . C. There are no horizontal asymptotes. nd any vertical asymptotes of \( f(x) \) Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, \( \square \) (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is \( \square \) and the rightmost asymptote is \( \square \) . (Type equations.) C. There are no vertical asymptotes.
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To summarize the pertinent information for the function \( f(x) = \frac{x+3}{x-3} \), we first analyze its horizontal and vertical asymptotes. As \( x \) approaches infinity, the highest degree terms dominate, leading to the horizontal asymptote at \( y = 1 \). So, the correct choice for horizontal asymptotes is A: The function has one horizontal asymptote, \( y = 1 \). Next, we determine the vertical asymptotes by identifying the values of \( x \) that make the denominator zero. Setting \( x - 3 = 0 \), we find a vertical asymptote at \( x = 3 \). Therefore, the correct choice for vertical asymptotes is A: The function has one vertical asymptote, \( x = 3 \). Now, you can easily sketch the graph—it's a hyperbola that approaches the horizontal asymptote (y=1) as x grows larger or smaller, and it will never touch the vertical asymptote at \( x=3 \). Happy graphing!
