Question

where are the vertical asymptotes for the following function located?

\\(f(x) = \frac{7}{x^2 - 2x - 24}\\)

\\(\bigcirc\\) \\(x = -4\\) and \\(x = 6\\)
\\(\bigcirc\\) \\(x = -4\\) and \\(x = 7\\)
\\(\bigcirc\\) \\(x = 4\\) and \\(x = -6\\)
\\(\bigcirc\\) \\(x = 6\\) and \\(x = 7\\)

Explanation:

Identify the condition for vertical asymptotes

$$ x^2 - 2x - 24 = 0 $$

Factor the quadratic denominator

$$ (x - 6)(x + 4) = 0 $$

Solve for the roots of the denominator

$$ x = 6 \quad \text{or} \quad x = -4 $$

Answer:

• (A) \(x = -4\) and \(x = 6\) (Correct answer)
• (B) \(x = -4\) and \(x = 7\)
• (C) \(x = 4\) and \(x = -6\)
• (D) \(x = 6\) and \(x = 7\)