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

Question

function.
h(x)=x+3/x(x-6)

Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Type an
equation. Use a comma 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 \( h(x) = \frac{x + 3}{x(x - 6)} \), we need to identify the vertical asymptotes and any holes in the graph.

1. Identifying Vertical Asymptotes: Vertical asymptotes occur where the denominator is zero and the numerator is not zero at those points.

   The denominator is \( x(x - 6) = 0 \) which can be factored to find the roots: \[ x = 0 \quad \text{or} \quad x - 6 = 0 \Rightarrow x = 6 \]

   We will check the numerator at these values:

   • For \( x = 0 \): \[ h(0) = \frac{0 + 3}{0(0 - 6)} = \text{undefined (denominator is zero)} \]
   • For \( x = 6 \): \[ h(6) = \frac{6 + 3}{6(6 - 6)} = \text{undefined (denominator is zero)} \]

   Both points \( x = 0 \) and \( x = 6 \) result in undefined values and are vertical asymptotes.

2. Identifying Holes: Holes occur if the same factor in the numerator and denominator cancels each other.

   The numerator \( x + 3 \) has no common factors with \( x \) or \( x - 6 \), thus there are no holes.

Putting it all together, our results are:

• Vertical asymptotes at \( x = 0 \) and \( x = 6 \).
• No holes.

Based on the choices given, the correct selection is:

A. The vertical asymptote(s) is(are) \( x = 0, x = 6 \) and hole(s) corresponding to none.