Find the domain of the function and identify any vertical and horizontal asymptotes.
$f ( x ) = \frac { x - 2 } { x ^ { 2 } - 4 }$
A)The domain is all real numbers x except x = ±2. There is a vertical asymptote at x = -2, and a horizontal asymptote at y = 0.
B)The domain is all real numbers x except x = ±2. There is a vertical asymptote at x = -2, and a horizontal asymptote at y = $\frac { 1 } { 2 }$ .
C)The domain is all real numbers x. There is a vertical asymptote at x = -4, and a horizontal asymptote at y = 0.
D)The domain is all real numbers x except x = ±4. There is a vertical asymptote at x = -4, and a horizontal asymptote at y = 0.
E)The domain is all real numbers x except x = ±4. There is a vertical asymptote at x = -2, and no horizontal asymptote.

Question

Find the domain of the function and identify any vertical and horizontal asymptotes. 
 $f ( x ) = \frac { x - 2 } { x ^ { 2 } - 4 }$  
A)The domain is all real numbers x except x = ±2. There is a vertical asymptote at x = -2, and a horizontal asymptote at y = 0.
B)The domain is all real numbers x except x = ±2. There is a vertical asymptote at x = -2, and a horizontal asymptote at y = $\frac { 1 } { 2 }$ .
C)The domain is all real numbers x. There is a vertical asymptote at x = -4, and a horizontal asymptote at y = 0.
D)The domain is all real numbers x except x  = ±4. There is a vertical asymptote at x = -4, and a horizontal asymptote at y = 0.
E)The domain is all real numbers x except x  = ±4. There is a vertical asymptote at x = -2, and no horizontal asymptote.

Quizplus · Accepted Answer

The domain excludes values that make the denominator zero, which are x = ±2, making the denominator $x^2 - 4 = (x-2)(x+2) = 0$. The function simplifies to $f(x) = \frac{1}{x+2}$ after canceling $x-2$ in numerator and denominator, leading to a vertical asymptote at x = -2. As $x$ approaches infinity, $f(x)$ approaches 0, indicating a horizontal asymptote at y = 0.

Find the Domain of the Function and Identify Any Vertical f(x)=x−2x2−4f ( x ) = \frac { x - 2 } { x ^ { 2 } - 4 }f(x)=x2−4x−2​

Find the Domain of the Function and Identify Any Vertical $f ( x ) = \frac { x - 2 } { x ^ { 2 } - 4 }$