Given The Function: F ( X ) = X 2 − 3 X − 4 2 X 2 − X − 10 F(x) = \frac{x^2 - 3x - 4}{2x^2 - X - 10} F ( X ) = 2 X 2 − X − 10 X 2 − 3 X − 4 ​ Analyze Or Solve As Required.

Introduction

In this article, we will delve into the analysis of a given rational function, $f(x) = \frac{x^2 - 3x - 4}{2x^2 - x - 10}$. Rational functions are a fundamental concept in mathematics, and understanding their properties is crucial for various applications in calculus, algebra, and other branches of mathematics. Our goal is to analyze and solve the given function, exploring its behavior, identifying its key features, and determining its domain and range.

Domain and Range

To begin with, let's examine the domain and range of the given function. The domain of a rational function consists of all real numbers except those that make the denominator zero. In this case, we need to find the values of $x$ that satisfy the equation $2x^2 - x - 10 = 0$. We can factor the quadratic expression as $(2x + 5)(x - 2) = 0$, which gives us two possible values for $x$: $x = -\frac{5}{2}$ and $x = 2$. Therefore, the domain of the function is all real numbers except $x = -\frac{5}{2}$ and $x = 2$.

import sympy as sp
x = sp.symbols('x')

denominator = 2*x**2 - x - 10

solutions = sp.solve(denominator, x)
print("The domain of the function is all real numbers except:")
for solution in solutions:
print(solution)

Simplifying the Function

Next, let's simplify the given function by factoring the numerator and denominator. We can factor the numerator as $(x - 4)(x + 1)$, and the denominator as $(2x + 5)(x - 2)$. Therefore, the simplified function is:

$$f(x) = \frac{(x - 4)(x + 1)}{(2x + 5)(x - 2)}$$

Identifying Key Features

Now that we have simplified the function, let's identify its key features. The function has two vertical asymptotes at $x = -\frac{5}{2}$ and $x = 2$, which are the values that make the denominator zero. Additionally, the function has a horizontal asymptote at $y = 0$, which is the ratio of the leading coefficients of the numerator and denominator.

import numpy as np

x_values = np.linspace(-10, 10, 400)

def f(x):
return (x - 4)(x + 1) / ((2x + 5)*(x - 2))

y_values = [f(x) for x in x_values]

print("The function has two vertical asymptotes at:")
print(-5/2)
print(2)
print("The function has a horizontal asymptote at:")
print(0)

Analyzing the Behavior

To analyze the behavior of the function, let's examine its graph. We can see that the function has two vertical asymptotes at $x = -\frac{5}{2}$ and $x = 2$, and a horizontal asymptote at $y = 0$. Additionally, the function has a hole at $x = -1$, which is the value that makes the numerator and denominator zero.

import matplotlib.pyplot as plt

x_values = np.linspace(-10, 10, 400)

def f(x):
return (x - 4)(x + 1) / ((2x + 5)*(x - 2))

y_values = [f(x) for x in x_values]

plt.plot(x_values, y_values)
plt.axvline(x=-5/2, color='red')
plt.axvline(x=2, color='red')
plt.axhline(y=0, color='green')
plt.scatter([-1], [f(-1)], color='blue')
plt.show()

Conclusion

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions related to the analysis of the given rational function, $f(x) = \frac{x^2 - 3x - 4}{2x^2 - x - 10}$.

Q: What is the domain of the function?

A: The domain of the function consists of all real numbers except those that make the denominator zero. In this case, we need to find the values of $x$ that satisfy the equation $2x^2 - x - 10 = 0$. We can factor the quadratic expression as $(2x + 5)(x - 2) = 0$, which gives us two possible values for $x$: $x = -\frac{5}{2}$ and $x = 2$. Therefore, the domain of the function is all real numbers except $x = -\frac{5}{2}$ and $x = 2$.

Q: What are the vertical asymptotes of the function?

A: The function has two vertical asymptotes at $x = -\frac{5}{2}$ and $x = 2$, which are the values that make the denominator zero.

Q: What is the horizontal asymptote of the function?

A: The function has a horizontal asymptote at $y = 0$, which is the ratio of the leading coefficients of the numerator and denominator.

Q: Is there a hole in the graph of the function?

A: Yes, there is a hole in the graph of the function at $x = -1$, which is the value that makes the numerator and denominator zero.

Q: How can I simplify the function?

A: We can simplify the function by factoring the numerator and denominator. We can factor the numerator as $(x - 4)(x + 1)$, and the denominator as $(2x + 5)(x - 2)$. Therefore, the simplified function is:

$$f(x) = \frac{(x - 4)(x + 1)}{(2x + 5)(x - 2)}$$

Q: How can I analyze the behavior of the function?

A: We can analyze the behavior of the function by examining its graph. We can see that the function has two vertical asymptotes at $x = -\frac{5}{2}$ and $x = 2$, and a horizontal asymptote at $y = 0$. Additionally, the function has a hole at $x = -1$, which is the value that makes the numerator and denominator zero.

Q: What is the importance of analyzing rational functions?

A: Analyzing rational functions is important because it helps us understand their behavior, identify their key features, and determine their domain and range. This knowledge is crucial for various applications in calculus, algebra, and other branches of mathematics.

Additional Resources

For further learning, we recommend the following resources:

• Textbooks: "Calculus" by Michael Spivak, "Algebra" by Michael Artin
• Online Courses: "Calculus" by MIT OpenCourseWare, "Algebra" by Khan Academy
• Software: Mathematica, Maple, Python

Conclusion

In conclusion, we have addressed some of the most frequently asked questions related to the analysis of the given rational function, $f(x) = \frac{x^2 - 3x - 4}{2x^2 - x - 10}$. We have provided detailed answers to each question, highlighting the importance of analyzing rational functions and providing additional resources for further learning.