Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( f(x)=\frac{x+3}{x-3} \) Find the intervals where \( f(x) \) is increasing or decreasing. Select the correct choice below and fill in the answer box(es) to complete your choice. (Type your answer in interval notation. Use a comma to separate answers as needed.) A. The function is increasing on B. The is never decreasing. C. The function is increasing on decreasing on \( \square \). It is decreasing never increasing. Find the location of any local extrema of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. There is a local minimum at \( x=\square \). There is no local maximum (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There is a local maximum at \( x=\square \) and there is a local minimum at \( x=\square \) (Type integers or decimals. Use a comma to separate answers as needed.) C. There is a local maximum at \( x=\square \). There is no local minimum. (Tvpe an integer or a decimal. Use a comma to separate answers as needed.)
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To analyze the function \( f(x) = \frac{x+3}{x-3} \), we start by finding its vertical asymptote. Setting the denominator equal to zero, we find that \( x = 3 \) is where the function is undefined. The horizontal asymptote can be determined by looking at the leading coefficients, which leads us to \( y = 1 \). The function increases as \( x \) approaches 3 from the left and approaches positive infinity. It then decreases as \( x \) moves from 3 to positive infinity. A. The function is increasing on \( (-\infty, 3) \) and decreasing on \( (3, \infty) \). For local extrema, we can find the derivative \( f'(x) \) to check for critical points. The function only has a vertical asymptote and doesn't have any other local extrema since it doesn't change from increasing to decreasing or vice versa aside from \( x = 3 \). C. There is a local maximum at \( x = 3 \). There is no local minimum.
