Question

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. f(x) = $$2x/(x^2 - 9)$$

Accepted Answer

Step 1: Identify the vertical asymptotes by finding the values of x that make the denominator equal to zero. Set the denominator $$x^2 - 9 = 0$$ and solve for x. This involves factoring the quadratic as $$(x - 3)(x + 3) = 0$$, leading to the solutions $$x = 3$$ and $$x = -3. $$These are the vertical asymptotes. Step 2: Determine the horizontal asymptote by comparing the degrees of the numerator and denominator. The numerator $$2x$$ has degree 1, and the denominator $$x^2 - 9$$ has degree 2. Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is $$y = 0. $$Step 3: Check for a slant asymptote. A slant asymptote exists if the degree of the numerator is exactly one greater than the degree of the denominator. In this case, the degree of the numerator is 1, and the degree of the denominator is 2, so there is no slant asymptote. Step 4: Summarize the asymptotes. The vertical asymptotes are $$x = 3$$ and $$x = -3$$, the horizontal asymptote is $$y = 0$$, and there is no slant asymptote. Step 5: To graph the function, plot the asymptotes and analyze the behavior of $$f(x)$$ near the asymptotes and at key points. For example, evaluate $$f(x) at $$values of $$x$$ near $$x = 3$$, $$x = -3$$, and $$x = 0 to $$understand how the function behaves.

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. f(x) = 2x/(x^2 - 9)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Vertical Asymptotes

Horizontal Asymptotes

Slant Asymptotes