Analyzing The Sign Of F(x) = (x-7)/(x^2 - 3x - 28) For X Less Than 7
In the realm of mathematical functions, understanding the behavior of a given function is paramount. This article delves into the intricacies of the function f(x) = (x-7)/(x^2 - 3x - 28), aiming to determine its positivity or negativity for values of x less than 7. To achieve this, we will employ a systematic approach, factoring the denominator, identifying critical points, and analyzing the function's sign in different intervals. Our journey will culminate in a definitive conclusion regarding the function's behavior for x < 7, providing a clear and concise answer to the posed question.
Factoring the Denominator: Unveiling the Roots
The first step in analyzing the function f(x) is to factor the denominator, which is a quadratic expression. Factoring the denominator will reveal the roots of the quadratic, which are crucial points where the function may change its sign. The denominator is given by x^2 - 3x - 28. To factor this quadratic, we seek two numbers that multiply to -28 and add up to -3. These numbers are -7 and 4. Therefore, we can factor the denominator as follows:
x^2 - 3x - 28 = (x - 7)(x + 4)
This factorization is a cornerstone for understanding the function's behavior. It tells us that the denominator becomes zero when x = 7 or x = -4. These values are critical points because they can potentially lead to changes in the function's sign. When the denominator is zero, the function is undefined, leading to vertical asymptotes or holes in the graph. Therefore, we must carefully consider these points when determining the intervals where the function is positive or negative.
Simplifying the Function: Identifying the True Nature
Now that we have factored the denominator, we can rewrite the function f(x) as follows:
f(x) = (x - 7) / [(x - 7)(x + 4)]
Notice that the factor (x - 7) appears in both the numerator and the denominator. This allows us to simplify the function, but with a crucial caveat. We can cancel the (x - 7) terms, but we must remember that x = 7 is not in the domain of the original function. Canceling the terms, we get:
f(x) = 1 / (x + 4), for x ≠ 7
This simplified form reveals the true nature of the function. It is a rational function with a vertical asymptote at x = -4 and a hole at x = 7. The simplification highlights that the behavior of the function is primarily determined by the (x + 4) term in the denominator. Understanding this simplified form is essential for analyzing the sign of the function.
Determining Critical Points: Mapping the Sign Changes
From the simplified form f(x) = 1 / (x + 4), we can identify the critical points that dictate the function's sign changes. A critical point is a value of x where the function is either zero or undefined. In this case, the function is undefined when the denominator is zero, which occurs at x = -4. There is no value of x that makes the numerator zero, so there are no zeros of the function.
Thus, the only critical point for the simplified function is x = -4. However, we must also consider the original function's domain. As we noted earlier, the original function is undefined at x = 7. Therefore, x = 7 is also a critical point, even though it doesn't appear explicitly in the simplified form. This point represents a hole in the graph of the function.
These critical points divide the number line into intervals. These intervals are (-∞, -4), (-4, 7), and (7, ∞). Within each of these intervals, the function's sign remains constant. To determine the sign of the function in each interval, we can test a value within the interval.
Analyzing the Sign in Intervals: Unveiling the Function's Behavior
Now, we analyze the sign of f(x) = 1 / (x + 4) in the intervals defined by the critical points x = -4 and x = 7. We will pick a test value within each interval and evaluate the function at that value.
- Interval (-∞, -4): Let's choose x = -5. Then, f(-5) = 1 / (-5 + 4) = 1 / (-1) = -1. Since f(-5) is negative, the function is negative in the interval (-∞, -4).
- Interval (-4, 7): Let's choose x = 0. Then, f(0) = 1 / (0 + 4) = 1 / 4. Since f(0) is positive, the function is positive in the interval (-4, 7).
- Interval (7, ∞): Let's choose x = 8. Then, f(8) = 1 / (8 + 4) = 1 / 12. Since f(8) is positive, the function is positive in the interval (7, ∞).
This analysis reveals a crucial insight. The function f(x) is negative in the interval (-∞, -4) and positive in the intervals (-4, 7) and (7, ∞). This information is vital for answering the question about the function's behavior for x < 7.
Conclusion: Determining the Function's Sign for x < 7
The original question asks whether f(x) is positive or negative for all x < 7. Based on our analysis, we know that f(x) is negative in the interval (-∞, -4) and positive in the interval (-4, 7). Therefore, it is not true that f(x) is positive for all x < 7, nor is it true that f(x) is negative for all x < 7.
f(x) takes on both positive and negative values for x < 7. Specifically, f(x) is negative when x < -4 and positive when -4 < x < 7. This detailed analysis provides a complete understanding of the function's behavior for the specified range of x values.
In conclusion, by factoring the denominator, simplifying the function, identifying critical points, and analyzing the sign in intervals, we have successfully determined the behavior of f(x) = (x-7)/(x^2 - 3x - 28) for x < 7. The function exhibits both positive and negative values in this range, highlighting the importance of a comprehensive analysis when studying mathematical functions.