Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.

Consider the following equation:
[tex]\[
f(x) = \frac{x^2 + 4}{4x^2 - 4x - 8}
\][/tex]

Name the vertical asymptote(s).

A. [tex]\( x = -1 \)[/tex] and [tex]\( x = 2 \)[/tex]

B. [tex]\( y = -1 \)[/tex] and [tex]\( y = 2 \)[/tex]

C. [tex]\( x = \frac{1}{4} \)[/tex]

D. [tex]\( y = \frac{1}{4} \)[/tex]

E. [tex]\( x = 0 \)[/tex]

F. [tex]\( y = 0 \)[/tex]


Sagot :

To determine the vertical asymptotes of the function

[tex]\[ f(x)=\frac{x^2+4}{4 x^2-4 x-8}, \][/tex]

we need to identify the values of [tex]\( x \)[/tex] that make the denominator equal to zero since these will cause the function to be undefined.

First, we start with the denominator:

[tex]\[ 4 x^2 - 4 x - 8 = 0. \][/tex]

This is a quadratic equation, and we can solve for [tex]\( x \)[/tex] using the quadratic formula:

[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \][/tex]

where [tex]\( a = 4 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = -8 \)[/tex].

First, calculate the discriminant:

[tex]\[ \text{Discriminant} = b^2 - 4ac = (-4)^2 - 4 \cdot 4 \cdot (-8) = 16 + 128 = 144. \][/tex]

Next, take the square root of the discriminant:

[tex]\[ \sqrt{\text{Discriminant}} = \sqrt{144} = 12. \][/tex]

Now, apply the quadratic formula:

[tex]\[ x = \frac{-(-4) \pm 12}{2 \cdot 4} = \frac{4 \pm 12}{8}. \][/tex]

This gives us two solutions:

[tex]\[ x_1 = \frac{4 + 12}{8} = \frac{16}{8} = 2, \][/tex]

[tex]\[ x_2 = \frac{4 - 12}{8} = \frac{-8}{8} = -1. \][/tex]

Hence, the vertical asymptotes of the function [tex]\( f(x) \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = -1 \)[/tex].

Thus, the correct answer is:

[tex]\[ \boxed{x=-1 \text{ and } x=2} \][/tex]