Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
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]
[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]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.