Analyzing The Sign Of F(x)=(x+4)/(x^2-3x-28)
In this article, we delve into the intricacies of determining the sign of a rational function. Specifically, we will analyze the function f(x) = (x+4)/(x^2-3x-28). Our goal is to understand where this function is positive, negative, or zero. This involves factoring, finding critical points, and testing intervals to assess the function's behavior. Understanding the sign of a function is crucial in various mathematical contexts, including solving inequalities, sketching graphs, and analyzing the behavior of systems.
Factoring and Identifying Critical Points
To begin our analysis, we need to factor the denominator of the rational function. The denominator is a quadratic expression, and factoring it will help us identify the points where the function is undefined (i.e., the vertical asymptotes) and the points where the function might change its sign. Factoring the quadratic x^2 - 3x - 28 involves finding two numbers that multiply to -28 and add to -3. These numbers are -7 and 4. Therefore, we can rewrite the denominator as (x - 7)(x + 4). Our function now looks like this:
f(x) = (x + 4) / ((x - 7)(x + 4))
Next, we identify the critical points of the function. These are the points where the function is either zero or undefined. The function is zero when the numerator is zero, which occurs when x = -4. The function is undefined when the denominator is zero, which occurs when x = 7 and x = -4. Notice that we have a common factor of (x + 4) in both the numerator and the denominator. This indicates a potential hole in the graph at x = -4. We can simplify the function by canceling out the common factor, but we must remember that x ≠-4.
After simplification, the function becomes:
f(x) = 1 / (x - 7), for x ≠-4
Now, we have two critical points to consider: x = -4 (where there's a hole) and x = 7 (where there's a vertical asymptote). These points divide the number line into three intervals: (-∞, -4), (-4, 7), and (7, ∞). We will test each interval to determine the sign of the function.
Analyzing the Intervals
To determine the sign of f(x) in each interval, we pick a test value within each interval and evaluate the function at that value. This will tell us whether the function is positive or negative in that interval.
- Interval (-∞, -4): Let's choose a test value of x = -5. Plugging this into the simplified function: f(-5) = 1 / (-5 - 7) = 1 / (-12) = -1/12 Since the result is negative, f(x) is negative in the interval (-∞, -4).
- Interval (-4, 7): Let's choose a test value of x = 0. Plugging this into the simplified function: f(0) = 1 / (0 - 7) = 1 / (-7) = -1/7 Since the result is negative, f(x) is negative in the interval (-4, 7).
- Interval (7, ∞): Let's choose a test value of x = 8. Plugging this into the simplified function: f(8) = 1 / (8 - 7) = 1 / (1) = 1 Since the result is positive, f(x) is positive in the interval (7, ∞).
From this analysis, we can see that f(x) is negative in the intervals (-∞, -4) and (-4, 7), and it is positive in the interval (7, ∞). This understanding of the intervals where the function is positive or negative is essential for solving inequalities and understanding the function's behavior graphically.
Evaluating the Given Statements
Now that we have a clear understanding of the sign of f(x) in different intervals, we can evaluate the given statements:
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A. f(x) is positive for all x > -4
This statement is false. We found that f(x) is negative in the interval (-4, 7) and positive only in the interval (7, ∞). Therefore, it is not positive for all x > -4.
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B. f(x) is negative for all x > -4
This statement is also false. As we determined, f(x) is negative in the interval (-4, 7) but positive in the interval (7, ∞). Thus, it is not negative for all x > -4.
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C. ... (The original text is incomplete, but we can infer that it's likely a statement about f(x) being zero or having a specific sign within a certain interval. Based on our analysis, we can deduce additional true or false statements)
To further explore, let's consider additional statements and analyze their validity based on our findings:
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D. f(x) is negative for -4 < x < 7
This statement is true. Our interval analysis clearly shows that f(x) is negative in the interval (-4, 7).
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E. f(x) is positive for x > 7
This statement is also true. Our analysis confirms that f(x) is positive in the interval (7, ∞).
By systematically analyzing the function, factoring, identifying critical points, and testing intervals, we can accurately determine the sign of the function in different regions and evaluate the truthfulness of various statements about its behavior. This comprehensive approach is essential for a thorough understanding of rational functions and their properties.
Graphical Interpretation
Visualizing the graph of the function f(x) = (x + 4) / (x^2 - 3x - 28) can provide further insights into its behavior. We know that there is a hole at x = -4 and a vertical asymptote at x = 7. The graph will approach the vertical asymptote as x gets closer to 7, and it will have a break (the hole) at x = -4. The simplified form of the function, f(x) = 1 / (x - 7), makes it clear that as x approaches 7 from the left, the function approaches negative infinity, and as x approaches 7 from the right, the function approaches positive infinity. This confirms our interval analysis.
The graph will be negative for x values between -4 and 7, which corresponds to the interval (-4, 7), and positive for x values greater than 7, corresponding to the interval (7, ∞). The hole at x = -4 is a point of discontinuity, but it does not change the overall sign of the function in the intervals around it. This graphical perspective reinforces our understanding of the function's sign and behavior.
Conclusion
Analyzing the sign of a rational function like f(x) = (x + 4) / (x^2 - 3x - 28) requires a systematic approach. This involves factoring the function, identifying critical points, testing intervals, and considering any discontinuities like holes or vertical asymptotes. By carefully examining these aspects, we can accurately determine where the function is positive, negative, or zero. This skill is valuable in various mathematical applications, including solving inequalities, sketching graphs, and understanding the behavior of functions in different contexts. Our detailed exploration has demonstrated how to break down a rational function and analyze its sign effectively, providing a solid foundation for further mathematical analysis.
In summary, understanding the behavior of rational functions is a critical skill in mathematics. By mastering techniques such as factoring, identifying critical points, and interval testing, we can confidently analyze and interpret the sign of these functions across different domains. This knowledge not only aids in problem-solving but also enhances our understanding of mathematical concepts and their applications in real-world scenarios. The ability to analyze functions effectively is a testament to mathematical proficiency and paves the way for more advanced studies in the field.
