Questions: Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of the rational function. f(x) = (x-3)/(x^2-9x+18)

Transcript text: Find the vertical asymptoles, if any, and the values of $x$ corresponding to holes, if any, of the graph of the rational function. \[ f(x)=\frac{x-3}{x^{2}-9 x+18} \]

Solution

Solution Steps

Step 1: Factor the Denominator

The given rational function is

\[ f(x) = \frac{x - 3}{x^2 - 9x + 18} \]

First, we factor the denominator:

\[ x^2 - 9x + 18 = (x - 6)(x - 3) \]

Step 2: Find the Roots of the Denominator

The roots of the denominator are found by setting each factor equal to zero:

\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \] \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

Step 3: Determine Vertical Asymptotes and Holes

To determine whether these roots correspond to vertical asymptotes or holes, we check the numerator at these points:

For $x = 3$, the numerator is $x - 3 = 0$. Since both the numerator and denominator are zero, there is a hole at $x = 3$.
For $x = 6$, the numerator is $x - 3 = 3$. Since the numerator is not zero, there is a vertical asymptote at $x = 6$.