Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of the rational

Question

function.

f(x)=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. The vertical asymptote(s) is(are) blank and hole(s) corresponding to blank
B. There are no vertical asymptotes but there is(are) hole(s) corresponding to blank.
C. The vertical asymptote(s) is(are) blank.
There are no holes.
D. There are no discontinuities.

Answer 1

To analyze the rational function \( f(x) = \frac{x}{x - 7} \), we need to identify any vertical asymptotes and holes.

1. Finding vertical asymptotes: Vertical asymptotes occur where the denominator is equal to zero, provided that the numerator is not also zero at those points.

   Set the denominator equal to zero: \[ x - 7 = 0 \implies x = 7 \] Since the numerator \( x \) does not equal zero when \( x = 7 \), we have a vertical asymptote at \( x = 7 \).

2. Finding holes: A hole occurs in the graph where both the numerator and the denominator are equal to zero. In this case, we check if there are any common factors in the numerator and denominator.

   The numerator is \( x \), which is equal to zero when: \[ x = 0 \]

   The denominator \( x - 7 \) is equal to zero when: \[ x - 7 = 0 \implies x = 7 \]

   Since \( x = 0 \) does not cancel with anything in the denominator, and there's no common factor that could create a hole, we conclude that there are no holes.

Thus, the conclusions are:

• The vertical asymptote is at \( x = 7 \).
• There are no holes.

Therefore, the correct choice is: C. The vertical asymptote(s) is(are) \( x = 7 \). There are no holes.