Question

Find all horizontal and vertical asymptotes. f(x) = 3x²2 - 16x + 5 x²-2x-15 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The horizontal asymptote(s) is/are (Type an equation. Use a comma to separate answers as needed.) B. The function has no horizontal asymptote

Answer 1

Given function is f(x) = 3x²2 - 16x + 5 / x²-2x-15To find the horizontal and vertical asymptotes of the function, we need to perform the following steps: Step 1: Factorize the denominator of the function. x²-2x-15 = (x-5)(x+3)Step 2: Rewrite the function in the form of a fraction with the factored denominator. f(x) = [3x²2 - 16x + 5] / [(x-5)(x+3)]Step 3: Check for the vertical asymptotes of the function by setting the denominator to zero and solving for x. x-5 = 0 or x+3 = 0 ⇒ x = 5 or x = -3Hence, the vertical asymptotes are x = 5 and x = -3.Step 4: Check for the horizontal asymptotes of the function by analyzing the degrees of the numerator and denominator. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.Therefore, the answer is:B. The function has no horizontal asymptote.

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