Given F ( X ) = 10 X 2 − 7 X − 30 F(x)=\frac{10}{x^2-7x-30} F ( X ) = X 2 − 7 X − 30 10 , Which Of The Following Is True?A. F ( X F(x F ( X ] Is Negative For All X \textless − 3 X \ \textless \ -3 X \textless − 3 .B. F ( X F(x F ( X ] Is Negative For All X \textgreater − 3 X \ \textgreater \ -3 X \textgreater − 3 .C. F ( X F(x F ( X ] Is Positive

Mar 8, 2025 by ADMIN 355 views

Analyzing the Behavior of Rational Functions: A Case Study

Rational functions are a fundamental concept in mathematics, and understanding their behavior is crucial for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will analyze the behavior of the rational function $f(x)=\frac{10}{x^2-7x-30}$ and determine which of the given statements is true.

Understanding Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In this case, the function $f(x)$ is a rational function with a numerator of 10 and a denominator of $x^2-7x-30$ . To analyze the behavior of this function, we need to understand the properties of rational functions.

Properties of Rational Functions

Rational functions have several properties that are essential for understanding their behavior. Some of these properties include:

Domain: The domain of a rational function is the set of all real numbers except those that make the denominator equal to zero.
Asymptotes: Rational functions can have vertical and horizontal asymptotes. Vertical asymptotes occur when the denominator is equal to zero, while horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator.
Sign of the function: The sign of a rational function can be determined by analyzing the signs of the numerator and denominator.

Analyzing the Denominator

To analyze the behavior of the function $f(x)$ , we need to examine the denominator $x^2-7x-30$ . This is a quadratic expression that can be factored as $(x-10)(x+3)$ . Therefore, the denominator is equal to zero when $x=10$ or $x=-3$ .

Determining the Sign of the Function

To determine the sign of the function $f(x)$ , we need to analyze the signs of the numerator and denominator. The numerator is a constant value of 10, which is positive. The denominator is a quadratic expression that is equal to zero when $x=10$ or $x=-3$ . Therefore, the sign of the function $f(x)$ is determined by the signs of the factors $(x-10)$ and $(x+3)$ .

Sign Analysis

To analyze the sign of the function $f(x)$ , we need to consider the intervals where the factors $(x-10)$ and $(x+3)$ are positive or negative.

Interval 1: $x<-3$ . In this interval, both factors $(x-10)$ and $(x+3)$ are negative. Therefore, the function $f(x)$ is positive.
Interval 2: $-3<x<10$ . In this interval, the factor $(x+3)$ is positive, while the factor $(x-10)$ is negative. Therefore, the function $f(x)$ is negative.
Interval 3: $x>10$ . In this interval, both factors $(x-10)$ and $(x+3)$ are positive. Therefore, the function $f(x)$ is positive.

Based on the analysis of the function $f(x)$ , we can conclude that:

Statement A is false. The function $f(x)$ is not negative for all $x<-3$ .
Statement B is false. The function $f(x)$ is not negative for all $x>-3$ .
Statement C is true. The function $f(x)$ is positive for $x<-3$ and $x>10$ .

In conclusion, analyzing the behavior of rational functions requires a deep understanding of their properties and the ability to analyze the signs of the numerator and denominator. By applying these concepts, we can determine the sign of the function $f(x)$ and conclude that statement C is true.

[1]: "Rational Functions" by Paul's Online Math Notes.
[2]: "Algebra and Trigonometry" by Michael Sullivan.

[1]: Khan Academy - Rational Functions.
[2]: MIT OpenCourseWare - Calculus.

What are some common mistakes that students make when analyzing rational functions? How can we improve our understanding of rational functions and their behavior? Share your thoughts and experiences in the comments below!
Frequently Asked Questions: Rational Functions

In our previous article, we analyzed the behavior of the rational function $f(x)=\frac{10}{x^2-7x-30}$ and determined which of the given statements is true. In this article, we will answer some frequently asked questions about rational functions and provide additional insights into their behavior.

Q: What is a rational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials. In other words, it is a function of the form $f(x)=\frac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials.

Q: What are the properties of rational functions?

A: Rational functions have several properties that are essential for understanding their behavior. Some of these properties include:

Domain: The domain of a rational function is the set of all real numbers except those that make the denominator equal to zero.
Asymptotes: Rational functions can have vertical and horizontal asymptotes. Vertical asymptotes occur when the denominator is equal to zero, while horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator.
Sign of the function: The sign of a rational function can be determined by analyzing the signs of the numerator and denominator.

Q: How do I determine the sign of a rational function?

A: To determine the sign of a rational function, you need to analyze the signs of the numerator and denominator. The sign of the function is determined by the signs of the factors in the numerator and denominator.

Q: What is the difference between a rational function and a polynomial?

A: A rational function is a function that can be expressed as the ratio of two polynomials, while a polynomial is a function that can be expressed as a sum of terms, each of which is a constant times a power of the variable.

Q: Can a rational function have a horizontal asymptote?

A: Yes, a rational function can have a horizontal asymptote if the degree of the numerator is less than or equal to the degree of the denominator.

Q: How do I find the vertical asymptotes of a rational function?

A: To find the vertical asymptotes of a rational function, you need to set the denominator equal to zero and solve for the variable.

Q: Can a rational function have a hole in its graph?

A: Yes, a rational function can have a hole in its graph if there is a factor in the numerator and denominator that cancels out.

Q: How do I determine the behavior of a rational function near a vertical asymptote?

A: To determine the behavior of a rational function near a vertical asymptote, you need to analyze the signs of the factors in the numerator and denominator.

In conclusion, rational functions are an important concept in mathematics, and understanding their behavior is crucial for solving various problems in algebra, calculus, and other branches of mathematics. By answering some frequently asked questions about rational functions, we hope to provide additional insights into their behavior and help you better understand this important concept.

[1]: "Rational Functions" by Paul's Online Math Notes.
[2]: "Algebra and Trigonometry" by Michael Sullivan.

[1]: Khan Academy - Rational Functions.
[2]: MIT OpenCourseWare - Calculus.