Question

where are the asymptotes for the following function located?
$f(x) = \frac{14}{(x - 5)(x + 1)}$
$x = 14$ and $x = 5$
$x = 1$ and $x = -5$
$x = -1$ and $x = 14$
$x = -1$ and $x = 5$

Explanation:

Step1: Recall Vertical Asymptote Rule

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), vertical asymptotes occur where \( D(x) = 0 \) (and \( N(x)
eq0 \) at those points).

Step2: Find Denominator Roots

Given \( f(x)=\frac{14}{(x - 5)(x + 1)} \), set denominator \( (x - 5)(x + 1)=0 \).
Solve \( x - 5 = 0 \) gives \( x = 5 \); solve \( x + 1 = 0 \) gives \( x=-1 \).

Answer:

\( x = -1 \) and \( x = 5 \) (the option with this text, likely the bottom - left or the one labeled with \( x=-1 \) and \( x = 5 \))