Given The Function:$\[ F(x) = \frac{12x^3 - 23x^2 + 24x - 11}{-4x^2 + 5x - 4} \\]Simplify Or Analyze The Function As Needed.

Introduction

In this article, we will be simplifying and analyzing the given function $f(x) = \frac{12x^3 - 23x^2 + 24x - 11}{-4x^2 + 5x - 4}$. The goal is to simplify the function as much as possible and analyze its behavior. We will use various mathematical techniques to achieve this goal.

Step 1: Factor the Numerator and Denominator

To simplify the function, we first need to factor the numerator and denominator. The numerator can be factored as follows:

${ 12x^3 - 23x^2 + 24x - 11 = (3x - 1)(4x^2 - 8x + 11) }$

The denominator can be factored as follows:

${ -4x^2 + 5x - 4 = -(4x^2 - 5x + 4) = -(4x - 1)(x - 4) }$

Step 2: Simplify the Function

Now that we have factored the numerator and denominator, we can simplify the function by canceling out any common factors. In this case, we can cancel out the factor $(3x - 1)$ from the numerator and denominator:

${ f(x) = \frac{(3x - 1)(4x^2 - 8x + 11)}{-(4x - 1)(x - 4)} = \frac{4x^2 - 8x + 11}{-(x - 4)} }$

Step 3: Simplify the Expression Further

We can simplify the expression further by factoring the numerator:

${ 4x^2 - 8x + 11 = (2x - 1)(2x - 11) }$

Now, we can rewrite the function as follows:

${ f(x) = \frac{(2x - 1)(2x - 11)}{-(x - 4)} }$

Step 4: Analyze the Function

Now that we have simplified the function, we can analyze its behavior. We can start by finding the domain of the function. The domain of the function is all real numbers except for the values of $x$ that make the denominator equal to zero. In this case, the denominator is equal to zero when $x = 4$. Therefore, the domain of the function is all real numbers except for $x = 4$.

Step 5: Find the Vertical Asymptote

The vertical asymptote of the function is the line $x = 4$. This is because the function is undefined at $x = 4$, and the graph of the function will approach the line $x = 4$ as $x$ approaches $4$ from either side.

Step 6: Find the Horizontal Asymptote

The horizontal asymptote of the function is the line $y = 0$. This is because the degree of the numerator is greater than the degree of the denominator, and the leading coefficient of the numerator is non-zero.

Conclusion

In this article, we simplified and analyzed the given function $f(x) = \frac{12x^3 - 23x^2 + 24x - 11}{-4x^2 + 5x - 4}$. We factored the numerator and denominator, canceled out common factors, and simplified the expression further. We also analyzed the function by finding its domain, vertical asymptote, and horizontal asymptote.

Final Simplified Function

The final simplified function is:

${ f(x) = \frac{(2x - 1)(2x - 11)}{-(x - 4)} }$

Graph of the Function

The graph of the function is a rational function with a vertical asymptote at $x = 4$ and a horizontal asymptote at $y = 0$. The graph will approach the line $x = 4$ as $x$ approaches $4$ from either side.

Real-World Applications

The given function has many real-world applications. For example, it can be used to model the growth of a population over time, where the numerator represents the rate of growth and the denominator represents the carrying capacity of the environment.

Limitations

The given function has some limitations. For example, it is undefined at $x = 4$, and the graph of the function will approach the line $x = 4$ as $x$ approaches $4$ from either side. This means that the function is not defined at $x = 4$, and the graph of the function will not be continuous at $x = 4$.

Future Research

There are many areas of future research related to the given function. For example, we can investigate the behavior of the function as $x$ approaches infinity or negative infinity. We can also investigate the behavior of the function as $x$ approaches the vertical asymptote at $x = 4$.

Conclusion

In conclusion, the given function $f(x) = \frac{12x^3 - 23x^2 + 24x - 11}{-4x^2 + 5x - 4}$ has been simplified and analyzed. We factored the numerator and denominator, canceled out common factors, and simplified the expression further. We also analyzed the function by finding its domain, vertical asymptote, and horizontal asymptote. The final simplified function is:

${ f(x) = \frac{(2x - 1)(2x - 11)}{-(x - 4)} }$

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Q: What is the final simplified function?

A: The final simplified function is:

${ f(x) = \frac{(2x - 1)(2x - 11)}{-(x - 4)} }$

Q: What is the domain of the function?

A: The domain of the function is all real numbers except for $x = 4$.

Q: What is the vertical asymptote of the function?

A: The vertical asymptote of the function is the line $x = 4$.

Q: What is the horizontal asymptote of the function?

A: The horizontal asymptote of the function is the line $y = 0$.

Q: What is the behavior of the function as $x$ approaches infinity or negative infinity?

A: As $x$ approaches infinity or negative infinity, the function approaches $y = 0$.

Q: What is the behavior of the function as $x$ approaches the vertical asymptote at $x = 4$?

A: As $x$ approaches the vertical asymptote at $x = 4$, the function approaches negative infinity.

Q: Can the function be used to model the growth of a population over time?

A: Yes, the function can be used to model the growth of a population over time, where the numerator represents the rate of growth and the denominator represents the carrying capacity of the environment.

Q: What are some limitations of the function?

A: Some limitations of the function include:

• The function is undefined at $x = 4$.
• The graph of the function will approach the line $x = 4$ as $x$ approaches $4$ from either side.
• The function is not defined at $x = 4$, and the graph of the function will not be continuous at $x = 4$.

Q: What are some areas of future research related to the function?

A: Some areas of future research related to the function include:

• Investigating the behavior of the function as $x$ approaches infinity or negative infinity.
• Investigating the behavior of the function as $x$ approaches the vertical asymptote at $x = 4$.
• Developing new mathematical models that can be used to describe the behavior of the function.

Q: How can the function be used in real-world applications?

A: The function can be used in a variety of real-world applications, including:

• Modeling the growth of a population over time.
• Analyzing the behavior of a system that is subject to a variety of constraints.
• Developing new mathematical models that can be used to describe the behavior of complex systems.

Q: What are some common mistakes that people make when simplifying and analyzing the function?

A: Some common mistakes that people make when simplifying and analyzing the function include:

• Failing to factor the numerator and denominator.
• Failing to cancel out common factors.
• Failing to analyze the behavior of the function as $x$ approaches infinity or negative infinity.
• Failing to analyze the behavior of the function as $x$ approaches the vertical asymptote at $x = 4$.

Q: How can people avoid making these mistakes?

A: People can avoid making these mistakes by:

• Carefully factoring the numerator and denominator.
• Carefully canceling out common factors.
• Analyzing the behavior of the function as $x$ approaches infinity or negative infinity.
• Analyzing the behavior of the function as $x$ approaches the vertical asymptote at $x = 4$.

Conclusion

In conclusion, the given function $f(x) = \frac{12x^3 - 23x^2 + 24x - 11}{-4x^2 + 5x - 4}$ has been simplified and analyzed. We have answered a variety of questions related to the function, including questions about its domain, vertical asymptote, horizontal asymptote, and behavior as $x$ approaches infinity or negative infinity. We have also discussed some areas of future research related to the function and some common mistakes that people make when simplifying and analyzing the function.