Analyzing F(x) = 10/(x^2 - 7x - 30) Determining Intervals Of Positivity And Negativity
In this article, we delve into the analysis of the sign of the rational function f(x) = 10/(x^2 - 7x - 30). Understanding where a function is positive or negative is a fundamental concept in mathematics, with applications ranging from solving inequalities to sketching graphs. We will explore the behavior of this function across different intervals of the x-axis by examining its critical points and the sign of its denominator. Specifically, we aim to determine the intervals where f(x) is negative or positive. This involves factoring the quadratic expression in the denominator, identifying the zeros, and analyzing the sign changes around these zeros. This comprehensive analysis will equip us with the ability to accurately determine the intervals where the function satisfies the given conditions.
Problem Statement
Given the function f(x) = 10/(x^2 - 7x - 30), we need to determine the intervals where the function is positive or negative. This requires us to analyze the behavior of the function across its domain, particularly around its critical points, which are the zeros of the denominator. The sign of a rational function is determined by the signs of its numerator and denominator. Since the numerator is a constant (10), which is always positive, the sign of f(x) depends solely on the sign of the denominator, x^2 - 7x - 30. Therefore, we need to find the roots of the quadratic equation x^2 - 7x - 30 = 0 and analyze the sign of the quadratic expression in the intervals defined by these roots. This involves factoring the quadratic expression, identifying the critical points, and then testing the sign of the function in each interval. This detailed approach will allow us to accurately determine the intervals where f(x) is negative or positive.
Factoring the Denominator
The first crucial step in analyzing the sign of f(x) is to factor the quadratic expression in the denominator, which is x^2 - 7x - 30. Factoring this quadratic will help us identify the values of x where the denominator is equal to zero, which are the points where the function is undefined. These points are critical because they divide the number line into intervals where the function's sign may change. To factor the quadratic, we are looking for two numbers that multiply to -30 and add up to -7. These numbers are -10 and 3. Therefore, we can write the quadratic expression as:
x^2 - 7x - 30 = (x - 10)(x + 3)
This factorization is essential as it allows us to easily find the roots of the quadratic equation x^2 - 7x - 30 = 0. By setting each factor equal to zero, we can find the values of x that make the denominator zero. These values will be the critical points around which we analyze the sign of the function. Accurate factorization is the cornerstone of this analysis, ensuring that we correctly identify the intervals where the function may change its sign.
Identifying Critical Points
Once we have factored the denominator as (x - 10)(x + 3), we can easily identify the critical points of the function f(x). Critical points are the values of x for which the denominator is equal to zero, as these are the points where the function is undefined. To find these points, we set each factor equal to zero:
- x - 10 = 0 => x = 10
- x + 3 = 0 => x = -3
These critical points, x = 10 and x = -3, are crucial because they divide the number line into three intervals: x < -3, -3 < x < 10, and x > 10. In each of these intervals, the sign of the denominator (x - 10)(x + 3) remains constant. This is because the quadratic expression can only change its sign at its roots. Therefore, we can analyze the sign of f(x) in each interval by choosing a test point within the interval and evaluating the sign of the denominator at that point. This step is essential for understanding the overall behavior of the function and determining where it is positive or negative.
Analyzing the Sign in Intervals
Now that we have identified the critical points x = -3 and x = 10, we can analyze the sign of the function f(x) = 10/(x^2 - 7x - 30) in the intervals they define. These intervals are:
- x < -3
- -3 < x < 10
- x > 10
To determine the sign of f(x) in each interval, we will choose a test value within the interval and evaluate the sign of the denominator (x - 10)(x + 3). Since the numerator is always positive (10), the sign of f(x) is solely determined by the sign of the denominator.
Interval 1: x < -3
Let's choose a test value, say x = -4. Plugging this into the denominator:
(-4 - 10)(-4 + 3) = (-14)(-1) = 14
The denominator is positive, so f(x) is positive in this interval.
Interval 2: -3 < x < 10
Let's choose a test value, say x = 0. Plugging this into the denominator:
(0 - 10)(0 + 3) = (-10)(3) = -30
The denominator is negative, so f(x) is negative in this interval.
Interval 3: x > 10
Let's choose a test value, say x = 11. Plugging this into the denominator:
(11 - 10)(11 + 3) = (1)(14) = 14
The denominator is positive, so f(x) is positive in this interval.
By analyzing the sign in each interval, we have a clear picture of the function's behavior across its domain. This detailed examination is crucial for answering specific questions about the function's positivity or negativity in various regions.
Conclusion
In conclusion, by analyzing the function f(x) = 10/(x^2 - 7x - 30), we have determined the intervals where the function is positive and negative. We started by factoring the denominator to find the critical points, which are x = -3 and x = 10. These points divide the number line into three intervals: x < -3, -3 < x < 10, and x > 10. By choosing test values within each interval, we evaluated the sign of the denominator and, consequently, the sign of the function.
Our analysis revealed the following:
- For x < -3, f(x) is positive.
- For -3 < x < 10, f(x) is negative.
- For x > 10, f(x) is positive.
This comprehensive analysis provides a clear understanding of the behavior of the rational function f(x). Such analyses are crucial in various mathematical contexts, including solving inequalities, sketching graphs, and understanding the properties of functions. The systematic approach we employed, from factoring the denominator to testing intervals, is a fundamental technique in the study of rational functions.
Now, let's address the specific statements:
A. f(x) is negative for all x < -3: This statement is false, as we found that f(x) is positive for x < -3. B. f(x) is negative for all x > -3: This statement is false, as f(x) is negative for -3 < x < 10, but positive for x > 10. C. f(x) is positive for all x > 10: This statement is true, as our analysis showed that f(x) is positive in the interval x > 10.
