Question

question 4 (1 point) saved
given the functions ( f(x) = x^2 - 9 ) and ( g(x) = x^2 - 5x - 6 ), determine the equations of the vertical asymptotes of the function ( y = \frac{f(x)}{g(x)} ).

a) ( x = 3, x = -3 )

b) ( x = 1, x = -6 )

c) ( x = -1, x = 6 )

d) ( x = 2, x = -3 )

Explanation:

Step1: Simplify the rational function

First, factor both \( f(x) \) and \( g(x) \).
\( f(x) = x^2 - 9 = (x - 3)(x + 3) \) (difference of squares).
\( g(x) = x^2 - 5x - 6 \). To factor \( g(x) \), find two numbers that multiply to \(-6\) and add to \(-5\). Those numbers are \(-6\) and \(1\), so \( g(x) = (x - 6)(x + 1) \).

The function \( y = \frac{f(x)}{g(x)} = \frac{(x - 3)(x + 3)}{(x - 6)(x + 1)} \). There are no common factors in the numerator and denominator, so we proceed to find vertical asymptotes.

Step2: Find vertical asymptotes

Vertical asymptotes occur where the denominator is zero (and the numerator is not zero, since there are no common factors).
Set the denominator equal to zero: \( (x - 6)(x + 1) = 0 \).
Solve for \( x \):
\( x - 6 = 0 \implies x = 6 \)
\( x + 1 = 0 \implies x = -1 \)

Answer:

c) \( x = -1, x = 6 \)