Questions: Find the vertical asymptote(s), if any, of the function f(x)=(x^2-5 x-24)/(x+3) x=8, x=-3 x=-3 x=1 x=8 There are no vertical asymptotes. None of the above

Solution Steps

To find the vertical asymptotes of a rational function, we need to identify 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 function \( f(x) = \frac{x^2 - 5x - 24}{x + 3} \), we set the denominator equal to zero and solve for \( x \).

To find the vertical asymptotes of the function \( f(x) = \frac{x^2 - 5x - 24}{x + 3} \), we first focus on the denominator, which is \( x + 3 \).

Set the denominator equal to zero and solve for \( x \): \[ x + 3 = 0 \] Solving this equation gives: \[ x = -3 \]

Ensure that the value \( x = -3 \) does not also make the numerator zero. The numerator is \( x^2 - 5x - 24 \). Substituting \( x = -3 \) into the numerator: \[ (-3)^2 - 5(-3) - 24 = 9 + 15 - 24 = 0 \] Since the numerator is zero when \( x = -3 \), this indicates a hole rather than a vertical asymptote.

Final Answer