Vertical Asymptotes Of F(x) = 2 / (3x(x-1)(x+5))

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In the realm of mathematical analysis, understanding the behavior of functions is paramount. One key aspect of this understanding is identifying vertical asymptotes. Vertical asymptotes are vertical lines that a function approaches but never quite reaches. They signify points where the function's value grows without bound, either positively or negatively. This article delves into the process of determining vertical asymptotes, focusing on the specific function F(x) = 2 / (3x(x-1)(x+5)). We will explore the underlying principles and apply them to identify the values of x at which this function exhibits vertical asymptotes.

Understanding Vertical Asymptotes

Vertical asymptotes of a function occur at x-values where the denominator of a rational function equals zero, while the numerator does not. This condition leads to an undefined value for the function, causing it to approach infinity (or negative infinity) as x approaches these specific values. In simpler terms, a vertical asymptote is like an insurmountable barrier for the function's graph. It gets infinitely close to the line but never crosses it.

To identify vertical asymptotes, we must first recognize the function as a rational function – a function expressed as the ratio of two polynomials. The function F(x) = 2 / (3x(x-1)(x+5)) clearly fits this description. The numerator is a constant (2), which can be considered a polynomial of degree zero, and the denominator is a polynomial of degree three, obtained by expanding 3x(x-1)(x+5). Once we have established that we are dealing with a rational function, the next step is to find the zeros of the denominator. These zeros are the potential locations of vertical asymptotes. However, it is crucial to verify that the numerator is non-zero at these points, as a zero in both the numerator and denominator might indicate a removable singularity (a hole in the graph) rather than a vertical asymptote.

Consider a general rational function of the form P(x) / Q(x), where P(x) and Q(x) are polynomials. The vertical asymptotes occur at the values of x for which Q(x) = 0, provided that P(x) ≠ 0 at those values. This is because division by zero is undefined in mathematics. As x gets closer and closer to a root of Q(x), the denominator approaches zero, and the value of the function grows without bound. This unbounded growth manifests as a vertical asymptote on the graph of the function. It's also important to note that the behavior of the function near a vertical asymptote can be different on the left and right sides of the asymptote. The function might approach positive infinity on one side and negative infinity on the other, or it might approach the same infinity (either positive or negative) on both sides. This behavior depends on the sign of the function as x approaches the asymptote from different directions.

Identifying Vertical Asymptotes for F(x) = 2 / (3x(x-1)(x+5))

Now, let's apply these principles to our function, F(x) = 2 / (3x(x-1)(x+5)). The first step is to identify the denominator, which is 3x(x-1)(x+5). To find the potential vertical asymptotes, we need to determine the values of x that make the denominator equal to zero. This involves solving the equation 3x(x-1)(x+5) = 0. The equation is already factored, which simplifies the process significantly.

The product of several factors is zero if and only if at least one of the factors is zero. Therefore, we can set each factor in the denominator equal to zero and solve for x:

  • 3x = 0
  • x - 1 = 0
  • x + 5 = 0

Solving these equations gives us the following values for x:

  • x = 0
  • x = 1
  • x = -5

These are the potential locations of vertical asymptotes. To confirm that they are indeed vertical asymptotes, we need to check that the numerator is non-zero at these points. The numerator of our function is 2, which is a constant and never equal to zero. Therefore, the values x = 0, x = 1, and x = -5 are confirmed to be vertical asymptotes of the function F(x).

Geometrically, this means that the graph of F(x) will have three vertical lines at x = -5, x = 0, and x = 1. As x approaches these values from either side, the function's value will tend towards positive or negative infinity. The lines act as barriers that the graph of the function cannot cross. The behavior of the function near these asymptotes can be visualized by plotting the graph or by analyzing the sign of the function in the intervals between the asymptotes. For instance, between x = -5 and x = 0, the function will either be positive or negative, and its value will increase or decrease without bound as it approaches the asymptotes. Similarly, the function's behavior can be analyzed in the intervals between x = 0 and x = 1, and to the left of x = -5 and to the right of x = 1.

Graphical Interpretation and Behavior Near Asymptotes

A visual representation of the function F(x) = 2 / (3x(x-1)(x+5)) can greatly enhance our understanding of vertical asymptotes. When we graph this function, we observe three distinct vertical lines at x = -5, x = 0, and x = 1. These lines represent the vertical asymptotes, and the graph of the function approaches these lines infinitely closely but never intersects them. The behavior of the function near these asymptotes is particularly interesting.

As x approaches -5 from the left (i.e., x < -5), the factors (x), (x-1), and (x+5) are all negative. Therefore, the denominator 3x(x-1)(x+5) is negative, and the function F(x) = 2 / (3x(x-1)(x+5)) is also negative. As x gets closer to -5, the factor (x+5) approaches zero, causing the denominator to become very small in magnitude and negative. Consequently, F(x) approaches negative infinity. On the other hand, as x approaches -5 from the right (i.e., x > -5), the factors (x) and (x-1) are still negative, but (x+5) is now positive. This makes the denominator positive, and F(x) becomes positive. As x gets closer to -5 from the right, F(x) approaches positive infinity. This change in sign as we cross the asymptote is a typical characteristic of vertical asymptotes.

A similar analysis can be performed for the asymptotes at x = 0 and x = 1. As x approaches 0 from the left, x is negative, (x-1) is negative, and (x+5) is positive. The denominator is thus positive, and F(x) is positive. As x approaches 0 from the left, F(x) approaches positive infinity. As x approaches 0 from the right, x is positive, (x-1) is negative, and (x+5) is positive. The denominator is negative, and F(x) is negative. As x approaches 0 from the right, F(x) approaches negative infinity. For the asymptote at x = 1, as x approaches 1 from the left, x is positive, (x-1) is negative, and (x+5) is positive. The denominator is negative, and F(x) is negative. As x approaches 1 from the left, F(x) approaches negative infinity. As x approaches 1 from the right, x is positive, (x-1) is positive, and (x+5) is positive. The denominator is positive, and F(x) is positive. As x approaches 1 from the right, F(x) approaches positive infinity.

These changes in sign and the unbounded growth or decay of the function near the asymptotes are visually apparent on the graph. The graph consists of several disconnected pieces, each bounded by the vertical asymptotes. The function's behavior between the asymptotes can also be analyzed. In the intervals (-∞, -5), (-5, 0), (0, 1), and (1, ∞), the function will be either positive or negative, and its value will vary continuously. The sign of the function in each interval can be determined by choosing a test value within the interval and evaluating F(x) at that value.

Conclusion

In summary, the function F(x) = 2 / (3x(x-1)(x+5)) has vertical asymptotes at x = -5, x = 0, and x = 1. These values are the zeros of the denominator of the rational function, and the numerator is non-zero at these points. Understanding vertical asymptotes is crucial for analyzing the behavior of rational functions and sketching their graphs. By identifying these asymptotes, we gain valuable insights into the function's behavior as x approaches certain values. The graphical interpretation of these asymptotes further enhances our understanding, showing how the function approaches infinity (or negative infinity) near these lines.

This exploration of vertical asymptotes for F(x) provides a solid foundation for analyzing more complex rational functions and their behavior. The principles discussed here can be applied to any rational function to identify its vertical asymptotes and understand its behavior near these points. The combination of algebraic techniques and graphical interpretation offers a powerful approach to mastering the intricacies of function analysis.

Answer

The values of x where the function F(x) has a vertical asymptote are:

  • A. -5
  • B. 0
  • F. 1