Analyzing F(x) = (9x - 6) / (-3x - 9) Domain, Asymptotes, And Behavior

Jul 10, 2025 by ADMIN 71 views

Exploring the Function f(x) = (9x - 6) / (-3x - 9): Domain, Asymptotes, and Behavior

Introduction

In this article, we delve into the analysis of the rational function f(x) = (9x - 6) / (-3x - 9). Our primary objective is to comprehensively understand the function's behavior by determining its domain, identifying its vertical and horizontal asymptotes, and examining its behavior in close proximity to the vertical asymptote. Rational functions, which are essentially ratios of two polynomials, exhibit fascinating properties, especially concerning their asymptotes and domains. Understanding these features is crucial in various fields, including calculus, real analysis, and engineering, as they provide insights into the function's limits, continuity, and overall nature. We will explore these concepts systematically, providing a clear and in-depth explanation of each aspect of the function. By the end of this discussion, you will have a solid grasp of how to analyze rational functions and interpret their key characteristics. This knowledge is not only beneficial for academic pursuits but also for practical applications where modeling and understanding functional behavior are paramount.

Determining the Domain

The domain of a function represents the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by any values of x that would make the denominator equal to zero, as division by zero is undefined. To find the domain of our function, f(x) = (9x - 6) / (-3x - 9), we need to identify the values of x that satisfy the equation -3x - 9 = 0. Solving this equation will reveal the x-values that must be excluded from the domain.

So, we set the denominator equal to zero:

-3x - 9 = 0

Adding 9 to both sides, we get:

-3x = 9

Dividing both sides by -3, we find:

x = -3

Therefore, the function is undefined when x = -3. This means that the domain of the function includes all real numbers except -3. In interval notation, the domain can be expressed as (-∞, -3) ∪ (-3, ∞). This notation indicates that the function is defined for all values of x less than -3 and all values of x greater than -3. The exclusion of -3 is critical for the function's definition, as it prevents the denominator from becoming zero, which would lead to an undefined expression. Understanding the domain is the first step in analyzing any function, as it sets the boundaries within which the function operates.

Identifying the Vertical Asymptote

A vertical asymptote occurs at a value of x where the function approaches infinity (or negative infinity). For rational functions, vertical asymptotes typically occur at the x-values that make the denominator zero, provided that these values do not also make the numerator zero. In our function, f(x) = (9x - 6) / (-3x - 9), we have already determined that the denominator is zero when x = -3. Now, we need to check if the numerator is also zero at x = -3.

The numerator is 9x - 6. Substituting x = -3 into the numerator, we get:

9(-3) - 6 = -27 - 6 = -33

Since the numerator is not zero at x = -3, we can conclude that there is a vertical asymptote at x = -3. The vertical asymptote is a vertical line that the function approaches but never touches. It represents a point of discontinuity in the function's graph. In this case, as x approaches -3, the function's value will either increase without bound (approach positive infinity) or decrease without bound (approach negative infinity). The behavior of the function near the vertical asymptote is crucial for understanding its overall shape and characteristics.

Analyzing the Behavior Near the Vertical Asymptote

To thoroughly analyze the behavior of the function f(x) = (9x - 6) / (-3x - 9) near the vertical asymptote at x = -3, we need to examine the function's limits as x approaches -3 from both the left (x → -3⁻) and the right (x → -3⁺). This will reveal how the function behaves as it gets infinitely close to the vertical asymptote.

Approaching from the Left (x → -3⁻)

When x approaches -3 from the left, it means x is slightly less than -3. Let's consider a value slightly less than -3, such as x = -3.01. Plugging this into the function, we get:

f(-3.01) = (9(-3.01) - 6) / (-3(-3.01) - 9) = (-27.09 - 6) / (9.03 - 9) = -33.09 / 0.03 = -1103

This result suggests that as x approaches -3 from the left, the function's value becomes a large negative number. We can confirm this by analyzing the signs of the numerator and denominator. As x approaches -3 from the left, the numerator (9x - 6) approaches 9(-3) - 6 = -33, which is negative. The denominator (-3x - 9) approaches -3(-3) - 9 = 0, but since x is slightly less than -3, -3x is slightly greater than 9, making the denominator a small positive number. Thus, we have a negative number divided by a small positive number, resulting in a large negative number. Mathematically, we can express this as:

lim (x→-3⁻) f(x) = -∞

Approaching from the Right (x → -3⁺)

When x approaches -3 from the right, it means x is slightly greater than -3. Let's consider a value slightly greater than -3, such as x = -2.99. Plugging this into the function, we get:

f(-2.99) = (9(-2.99) - 6) / (-3(-2.99) - 9) = (-26.91 - 6) / (8.97 - 9) = -32.91 / -0.03 = 1097

This result suggests that as x approaches -3 from the right, the function's value becomes a large positive number. Analyzing the signs again, as x approaches -3 from the right, the numerator (9x - 6) still approaches -33, which is negative. However, the denominator (-3x - 9) approaches 0, but since x is slightly greater than -3, -3x is slightly less than 9, making the denominator a small negative number. Thus, we have a negative number divided by a small negative number, resulting in a large positive number. Mathematically, we can express this as:

lim (x→-3⁺) f(x) = ∞

Conclusion

In conclusion, as x approaches the vertical asymptote x = -3 from the left, the function decreases without bound, approaching negative infinity. Conversely, as x approaches -3 from the right, the function increases without bound, approaching positive infinity. This behavior is characteristic of rational functions near their vertical asymptotes and provides valuable insights into the function's graph and overall behavior. Understanding these limits is crucial for sketching the graph of the function and for applications in calculus and analysis.

Determining the Horizontal Asymptote

A horizontal asymptote describes the behavior of the function as x approaches positive or negative infinity. It is a horizontal line that the function approaches but may or may not cross. To find the horizontal asymptote of our function, f(x) = (9x - 6) / (-3x - 9), we need to examine the limits of the function as x approaches infinity and negative infinity.

Limit as x Approaches Infinity (x → ∞)

To find the limit as x approaches infinity, we divide both the numerator and the denominator by the highest power of x present in the function, which in this case is x:

lim (x→∞) f(x) = lim (x→∞) [(9x - 6) / x] / [(-3x - 9) / x]

This simplifies to:

lim (x→∞) (9 - 6/x) / (-3 - 9/x)

As x approaches infinity, the terms 6/x and 9/x approach 0. Therefore, the limit becomes:

lim (x→∞) (9 - 0) / (-3 - 0) = 9 / -3 = -3

Limit as x Approaches Negative Infinity (x → -∞)

Similarly, to find the limit as x approaches negative infinity, we perform the same steps:

lim (x→-∞) f(x) = lim (x→-∞) [(9x - 6) / x] / [(-3x - 9) / x]

This simplifies to:

lim (x→-∞) (9 - 6/x) / (-3 - 9/x)

As x approaches negative infinity, the terms 6/x and 9/x still approach 0. Therefore, the limit becomes:

lim (x→-∞) (9 - 0) / (-3 - 0) = 9 / -3 = -3

Conclusion

Both limits, as x approaches infinity and negative infinity, are equal to -3. This indicates that the function has a horizontal asymptote at y = -3. The horizontal asymptote provides insight into the function's long-term behavior, showing that as x moves further away from zero in either direction, the function's value gets closer and closer to -3. This asymptote serves as a guide for sketching the graph of the function and is a crucial characteristic for understanding the function's behavior over its entire domain.

Conclusion

In summary, for the function f(x) = (9x - 6) / (-3x - 9), we have determined the following key characteristics:

Domain: The domain of the function is all real numbers except x = -3, expressed in interval notation as (-∞, -3) ∪ (-3, ∞).
Vertical Asymptote: There is a vertical asymptote at x = -3.
Behavior Near Vertical Asymptote: As x approaches -3 from the left, f(x) approaches negative infinity. As x approaches -3 from the right, f(x) approaches positive infinity.
Horizontal Asymptote: There is a horizontal asymptote at y = -3.

Understanding these aspects allows for a comprehensive analysis of the function's behavior and graphical representation. The domain restrictions dictate where the function is defined, while the asymptotes provide crucial information about the function's limits and long-term behavior. The behavior near the vertical asymptote further refines our understanding of the function's discontinuities and how it behaves in critical regions. This analysis is not only essential for academic understanding but also for various applications in fields such as engineering, physics, and economics, where rational functions are used to model real-world phenomena. By mastering the techniques to analyze functions like this, one gains a valuable tool for problem-solving and critical thinking in a wide array of disciplines.