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Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of \( \mathrm{y}=\mathrm{f}(\mathrm{x}) \). \( \mathrm{f}(\mathrm{x})=\frac{x^{2}+10 \mathrm{x}+24}{x^{2}+8 \mathrm{x}+16} \) Find any oblique asymptotes of \( \mathrm{f}(\mathrm{x}) \). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one oblique asymptote, (Type an equation.) B. The function has two different oblique asymptotes. The oblique asymptote as x approaches - \( \infty \) is (Type equations.) C. There are no oblique asymptotes. Find the oblique asymptote as \( x \) approaches \( \infty \) is A. The function is increasing on \( \square \). It is never decreasing. (Type an exact answer. Type your answer in interval notation. Use a comma to separate answers as needed.) B. The function is increasing on \( \square \). It is decreasing on \( \square \). (Type exact answers. Type your answers in interval notation. Use commas to separate answers as needed.) C. The function is decreasing on \( \square \). It is never increasing. Tvne an exart answer Tvne vour answer in interval notation Isse a romma to senarate answers as needed)

Ask by Pearson Williams. in the United States
Mar 17,2025

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Answer

There are no oblique asymptotes. The function is decreasing on \( (-\infty, -4) \) and \( (-4, \infty) \).

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The Deep Dive

To analyze the function \( f(x) = \frac{x^2 + 10x + 24}{x^2 + 8x + 16} \), we first simplify the expression. The denominator factors to \( (x + 4)^2 \) while the numerator factors to \( (x + 6)(x + 4) \). This allows us to simplify \( f(x) = \frac{x + 6}{x + 4} \) for \( x \neq -4 \). As \( x \) approaches \( \infty \) or \( -\infty \), \( f(x) \) approaches \( 1 \). Therefore, there are no oblique asymptotes, confirming option C: There are no oblique asymptotes. The function increases on \( (-\infty, -4) \) and \( (-4, \infty) \) due to the absence of critical points derived from \( f'(x) \). Thus, option A is the appropriate selection since the function never decreases. Summarizing: There are no oblique asymptotes for \( f(x) = \frac{x^2 + 10x + 24}{x^2 + 8x + 16} \), and the function is increasing on the intervals \( (-\infty, -4) \) and \( (-4, \infty) \). The graph of \( f(x) \) will show a horizontal asymptote at \( y = 1 \), and a hole at \( x = -4 \). Enjoy sketching your function with these elements in mind!

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