Questions: Find the vertical asymptotes, if any, and f(x)=(x-8)/(x^2-11x+24)

Find the vertical asymptotes, if any, and f(x)=(x-8)/(x^2-11x+24)
Transcript text: Find the vertical asymptotes, if any, anc \[ f(x)=\frac{x-8}{x^{2}-11 x+24} \]
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Solution

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Solution Steps

To find the vertical asymptotes of a rational function, we need to determine the values of \( x \) that make the denominator zero, as long as these values do not also make the numerator zero (which would indicate a hole instead of an asymptote). For the given function \( f(x) = \frac{x-8}{x^2 - 11x + 24} \), we will solve the equation \( x^2 - 11x + 24 = 0 \) to find the potential vertical asymptotes.

Step 1: Identify the Function

We are given the function

\[ f(x) = \frac{x-8}{x^2 - 11x + 24}. \]

Step 2: Set the Denominator to Zero

To find the vertical asymptotes, we need to solve the equation

\[ x^2 - 11x + 24 = 0. \]

Step 3: Factor the Quadratic

Factoring the quadratic, we find:

\[ (x - 3)(x - 8) = 0. \]

Step 4: Solve for \( x \)

Setting each factor to zero gives us the potential vertical asymptotes:

\[ x - 3 = 0 \quad \Rightarrow \quad x = 3, \] \[ x - 8 = 0 \quad \Rightarrow \quad x = 8. \]

Step 5: Determine Vertical Asymptotes

Since neither \( x = 3 \) nor \( x = 8 \) makes the numerator \( x - 8 \) zero, both values are indeed vertical asymptotes.

Final Answer

The vertical asymptotes are

\[ \boxed{x = 3} \quad \text{and} \quad \boxed{x = 8}. \]

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