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Math  /  Algebra

QuestionFind the vertical asymptotes, if any, and the values of xx corresponding to holes, if any, of the graph of the rational function. f(x)=xx+7f(x)=\frac{x}{x+7}
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Type an equation. Use commas to separate answers as needed.) A. There are no vertical asymptotes but there is(are) hole(s) corresponding to □\square . B. The vertical asymptote(s) is(are) □\square . There are no holes. C. The vertical asymptote(s) is(are) □\square and hole(s) corresponding to □\square . D. There are no discontinuities.

Studdy Solution

STEP 1

1. A vertical asymptote occurs where the denominator of a rational function is zero and the numerator is not zero at that point.
2. A hole occurs where both the numerator and the denominator are zero at the same point, indicating a removable discontinuity.
3. We need to analyze the function f(x)=xx+7 f(x) = \frac{x}{x+7} to determine the points of discontinuity.

STEP 2

1. Identify points where the denominator is zero.
2. Determine if these points are vertical asymptotes or holes.
3. Select the correct choice based on the analysis.

STEP 3

Identify points where the denominator is zero:
The denominator of f(x)=xx+7 f(x) = \frac{x}{x+7} is x+7 x+7 .
Set the denominator equal to zero to find the points of discontinuity:
x+7=0 x + 7 = 0
Solve for x x :
x=−7 x = -7

STEP 4

Determine if the point x=−7 x = -7 is a vertical asymptote or a hole:
1. Check the numerator at x=−7 x = -7 :
The numerator is x x , so at x=−7 x = -7 , the numerator is −7-7, which is not zero.
2. Since the numerator is not zero at x=−7 x = -7 , this point is a vertical asymptote, not a hole.

STEP 5

Select the correct choice based on the analysis:
Since there is a vertical asymptote at x=−7 x = -7 and no holes, the correct choice is:
B. The vertical asymptote(s) is(are) x=−7 x = -7 . There are no holes.
The vertical asymptote is x=−7 x = -7 and there are no holes.

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