Question

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

Accepted Answer

Step 1: Identify the vertical asymptotes by looking for values of x where the function approaches infinity or negative infinity. In the graph, these are the vertical dashed lines where the function is undefined. Here, the vertical asymptotes are at $$x = -4 ($$red dashed line) and $$x = 3 ($$orange dashed line). Step 2: Identify the horizontal asymptote by observing the behavior of the function as $$x$$ approaches positive or negative infinity. The graph shows the function approaching the horizontal dashed purple line $$y = -2 as$$ $$x$$ goes to both $$+\infty$$ and $$-\infty. So$$, the horizontal asymptote is $$y = -2. $$Step 3: Check for any oblique (slant) asymptotes by looking for a line that the graph approaches as $$x$$ goes to infinity, which is neither horizontal nor vertical. In this graph, there is no oblique asymptote since the function approaches a horizontal line instead. Step 4: State the domain of the function $$f. $$The domain includes all real numbers except where the vertical asymptotes occur, because the function is undefined at those points. Therefore, the domain is all real numbers except $$x = -4$$ and $$x = 3. $$Step 5: Summarize the asymptotes and domain: Vertical asymptotes at $$x = -4$$ and $$x = 3$$, horizontal asymptote at $$y = -2$$, no oblique asymptote, and domain $$(-\infty, -4) \cup (-4, 3) \cup (3, \infty).$$

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Vertical Asymptotes

Horizontal and Oblique Asymptotes

Domain of a Function