Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine the vertical asymptotes of the function
[tex]\[ f(x) = \frac{x^5 + 4x^3 - 4x + 6}{x^3 - 2x^2 - 5x - 6}, \][/tex]
we need to find where the denominator equals zero, as these points cause the function to be undefined and thus might indicate vertical asymptotes.
### Step-by-Step Solution:
1. Identify the Denominator:
The denominator of the function is:
[tex]\[ x^3 - 2x^2 - 5x - 6 \][/tex]
2. Set the Denominator Equal to Zero:
We need to find the roots of the equation:
[tex]\[ x^3 - 2x^2 - 5x - 6 = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
Solve the cubic equation for [tex]\( x \)[/tex]:
[tex]\[ x = 2/3 + (-1/2 - \sqrt{3} i/2) (\sqrt{137}/3 + 134/27)^{1/3} + 19/(9(-1/2 - \sqrt{3} i/2)(\sqrt{137}/3 + 134/27)^{1/3}), \][/tex]
[tex]\[ x = 2/3 + (-1/2 + \sqrt{3} i/2) (\sqrt{137}/3 + 134/27)^{1/3} + 19/(9(-1/2 + \sqrt{3} i/2)(\sqrt{137}/3 + 134/27)^{1/3}), \][/tex]
[tex]\[ x = 2/3 + 19/(9 (\sqrt{137}/3 + 134/27)^{1/3}) + (\sqrt{137}/3 + 134/27)^{1/3} \][/tex]
These are the roots of the denominator. Because the roots are complex numbers, they do not correspond to vertical asymptotes in the real number plane.
### Conclusion:
Since all the roots of the denominator are complex, there are no vertical asymptotes for the function [tex]\( f(x) \)[/tex] in the real number plane.
[tex]\[ f(x) = \frac{x^5 + 4x^3 - 4x + 6}{x^3 - 2x^2 - 5x - 6}, \][/tex]
we need to find where the denominator equals zero, as these points cause the function to be undefined and thus might indicate vertical asymptotes.
### Step-by-Step Solution:
1. Identify the Denominator:
The denominator of the function is:
[tex]\[ x^3 - 2x^2 - 5x - 6 \][/tex]
2. Set the Denominator Equal to Zero:
We need to find the roots of the equation:
[tex]\[ x^3 - 2x^2 - 5x - 6 = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
Solve the cubic equation for [tex]\( x \)[/tex]:
[tex]\[ x = 2/3 + (-1/2 - \sqrt{3} i/2) (\sqrt{137}/3 + 134/27)^{1/3} + 19/(9(-1/2 - \sqrt{3} i/2)(\sqrt{137}/3 + 134/27)^{1/3}), \][/tex]
[tex]\[ x = 2/3 + (-1/2 + \sqrt{3} i/2) (\sqrt{137}/3 + 134/27)^{1/3} + 19/(9(-1/2 + \sqrt{3} i/2)(\sqrt{137}/3 + 134/27)^{1/3}), \][/tex]
[tex]\[ x = 2/3 + 19/(9 (\sqrt{137}/3 + 134/27)^{1/3}) + (\sqrt{137}/3 + 134/27)^{1/3} \][/tex]
These are the roots of the denominator. Because the roots are complex numbers, they do not correspond to vertical asymptotes in the real number plane.
### Conclusion:
Since all the roots of the denominator are complex, there are no vertical asymptotes for the function [tex]\( f(x) \)[/tex] in the real number plane.
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.