Analyzing The Behavior Of Rational Function F(x) = (2x^4 + 9) / (3x^2 - 7)

When analyzing rational functions, such as the given function f(x) = (2x^4 + 9) / (3x^2 - 7), it's essential to understand their behavior as x approaches positive and negative infinity. This involves examining the leading terms of the numerator and the denominator. In this case, the leading term in the numerator is 2x^4, and the leading term in the denominator is 3x^2. As x becomes very large (either positively or negatively), these leading terms will dominate the function's behavior.

To analyze the end behavior, we can divide the leading terms: (2x^4) / (3x^2) = (2/3)x^2. This simplified expression tells us a great deal about what happens to f(x) as x moves towards infinity. Since x is squared, the result will always be positive, irrespective of whether x is a large positive number or a large negative number. This implies that as x moves further away from zero in either direction, f(x) will increase without bound.

However, a comprehensive understanding requires more than just looking at end behavior. We must also consider the function's vertical asymptotes. These occur where the denominator of the rational function equals zero. For our function, this happens when 3x^2 - 7 = 0. Solving for x, we get:

3x^2 = 7

x^2 = 7/3

x = ±√(7/3)

So, we have vertical asymptotes at x = √(7/3) and x = -√(7/3). These asymptotes play a crucial role in the function's behavior because as x approaches these values, the function will tend towards positive or negative infinity. Specifically, we need to analyze the behavior on either side of these asymptotes to understand how the function behaves in these regions.

To the left of x = -√(7/3), both the numerator and the denominator are positive. As x approaches −∞, the function increases without bound, mirroring the end behavior we discussed earlier. As x approaches −√(7/3) from the left, the denominator approaches zero from the positive side, and hence, the function increases towards positive infinity.

Between the two asymptotes, let's pick a test value, say x = 0. At x = 0, f(x) = (2(0)^4 + 9) / (3(0)^2 - 7) = 9 / -7, which is negative. This means that between x = -√(7/3) and x = √(7/3), the function is negative. As x approaches −√(7/3) from the right, the denominator approaches zero from the negative side, making the function approach negative infinity. Similarly, as x approaches √(7/3) from the left, the function approaches negative infinity.

To the right of x = √(7/3), both the numerator and the denominator are positive again. As x approaches √(7/3) from the right, the denominator approaches zero from the positive side, and hence, the function increases towards positive infinity. As x goes to +∞, the function continues to increase without bound, consistent with our analysis of the end behavior.

Therefore, the rational function f(x) = (2x^4 + 9) / (3x^2 - 7) increases without bound as x decreases without bound and as x increases without bound. Understanding the interplay between end behavior and vertical asymptotes gives a clear picture of the function's overall dynamics.

To fully grasp the behavior of the rational function f(x) = (2x^4 + 9) / (3x^2 - 7), a comprehensive analysis is necessary. This analysis not only involves considering the end behavior and vertical asymptotes but also delving into the function's symmetry, intercepts, and local extrema. By examining these different aspects, we can develop a precise understanding of how the function behaves across its domain.

Firstly, let's revisit the end behavior. As previously discussed, the leading terms 2x^4 in the numerator and 3x^2 in the denominator dictate the function's trend as x approaches infinity. The ratio of these terms simplifies to (2/3)x^2, a quadratic expression that opens upwards. Thus, as x approaches either positive or negative infinity, f(x) increases without bound. This provides a general idea of the function's long-term behavior, suggesting that the function will trend upwards at both ends of its domain.

Next, the vertical asymptotes are critical features of the rational function. They occur where the denominator equals zero, which we found to be at x = ±√(7/3). These vertical lines act as barriers that the function cannot cross. The function approaches infinity (positive or negative) as x gets closer to these values, making the asymptotes essential reference points for sketching the graph and understanding the function's local behavior. To determine the direction of approach (positive or negative infinity), we analyzed the sign of the function in intervals separated by the asymptotes. This revealed that the function goes to positive infinity as it approaches each asymptote from the sides that cause the denominator to approach zero from the positive side, and to negative infinity as it approaches from the sides where the denominator approaches zero from the negative side.

The y-intercept is another crucial point to consider. It's where the function intersects the y-axis, which occurs when x = 0. Substituting x = 0 into the function, we get:

f(0) = (2(0)^4 + 9) / (3(0)^2 - 7) = 9 / -7 = -9/7

So the y-intercept is at (0, -9/7). This point gives us a specific location on the graph, helping anchor the curve's position on the coordinate plane. There is no x-intercept, because the numerator 2x^4 + 9 is always positive and never equal to zero.

Another important consideration is the symmetry of the function. A function is even if f(x) = f(-x) for all x in its domain, and odd if f(x) = -f(-x). In this case,

f(-x) = (2(-x)^4 + 9) / (3(-x)^2 - 7) = (2x^4 + 9) / (3x^2 - 7) = f(x)

Since f(x) = f(-x), the function is even. This means the graph of the function is symmetric about the y-axis, simplifying our analysis as we only need to fully investigate the function’s behavior on one side of the y-axis.

To fully understand the local behavior, we would typically find local extrema by taking the derivative of the function and finding its critical points. However, for the scope of this discussion, we emphasize that the asymptotic behavior significantly influences the function's shape. As the function approaches its asymptotes and extends to infinity, the behavior around the asymptotes and the end behavior will determine the local behavior to some extent.

In summary, the comprehensive analysis of the rational function f(x) = (2x^4 + 9) / (3x^2 - 7) involves considering its end behavior, vertical asymptotes, intercepts, and symmetry. The end behavior reveals that the function increases without bound as x approaches positive or negative infinity. The vertical asymptotes at x = ±√(7/3) are boundaries the function approaches but never crosses. The y-intercept is at (0, -9/7), and the function’s even symmetry simplifies the analysis. Combining these factors gives a detailed picture of how the function behaves across its domain.

Visualizing the rational function f(x) = (2x^4 + 9) / (3x^2 - 7) graphically provides a deeper understanding of its behavior. The graph encapsulates all the analytical elements discussed earlier, including end behavior, vertical asymptotes, intercepts, and symmetry, allowing us to see how these features interact to shape the function.

The graph confirms that as x moves towards positive or negative infinity, the function f(x) increases without bound. This upward trend is consistent with our analysis of the end behavior, where the dominant term (2/3)x^2 dictates the function’s long-term path. The quadratic nature of this term ensures that f(x) will increase in both directions, making the ends of the graph point upwards.

The vertical asymptotes at x = √(7/3) and x = -√(7/3) are clearly visible on the graph as vertical lines. The function approaches these lines but never crosses them. On either side of each asymptote, the function moves towards positive or negative infinity, showcasing the dramatic effect these asymptotes have on the function’s local behavior. Specifically, as x approaches the asymptotes from the sides where the denominator (3x^2 - 7) approaches zero from the positive side, f(x) tends to positive infinity. Conversely, as x approaches the asymptotes from the sides where the denominator approaches zero from the negative side, f(x) tends to negative infinity. This behavior is crucial in understanding how the function behaves in the vicinity of these critical points.

The y-intercept at (0, -9/7) is another key feature that can be readily identified on the graph. This point anchors the function on the y-axis and provides a reference for the function’s vertical positioning. The fact that there are no x-intercepts is also visually confirmed, as the curve never crosses the x-axis.

The symmetry of the function about the y-axis is evident in the graph’s mirror-image appearance on either side of the y-axis. This symmetry, stemming from the function’s even nature, simplifies our analysis and visualization, as the behavior on one side of the y-axis is mirrored on the other side.

Beyond these features, the graph also highlights the local minima of the function between the vertical asymptotes and the end behavior. These are the turning points where the function changes direction. A full analytical determination of these points would involve calculus, but their presence is apparent on the graph, adding further detail to our understanding of the function’s behavior.

In summary, the graphical representation of f(x) = (2x^4 + 9) / (3x^2 - 7) is a comprehensive tool for understanding its characteristics. The graph visually confirms the function’s end behavior, vertical asymptotes, intercept, and symmetry, and even hints at its local minima. It pulls together the different aspects of the function’s behavior into a single, coherent picture, providing an intuitive grasp of how the function behaves across its domain. This graphical approach enhances the analytical understanding, making it easier to grasp the dynamics of the rational function.

In conclusion, the rational function f(x) = (2x^4 + 9) / (3x^2 - 7) exhibits rich and varied behavior that can be thoroughly understood through a combination of analytical and graphical techniques. The function’s end behavior, dictated by the ratio of the leading terms, results in f(x) increasing without bound as x approaches positive or negative infinity. This characteristic is crucial in outlining the function’s long-term trend.

The presence of vertical asymptotes at x = √(7/3) and x = -√(7/3) significantly influences the local behavior of the function. These asymptotes serve as boundaries that the function approaches but never crosses, causing f(x) to tend towards positive or negative infinity as x gets closer to these critical values. The detailed analysis around the asymptotes is essential for grasping the function's local dynamics.

The y-intercept at (0, -9/7) provides a specific anchor point on the graph, helping to position the curve accurately. The absence of x-intercepts, due to the numerator being always positive, is another key characteristic that shapes the function’s appearance.

The function’s even symmetry, confirmed both analytically and graphically, simplifies the analysis by allowing us to focus primarily on one side of the y-axis. This symmetry underscores the balance and predictability of the function’s behavior.

Ultimately, a graphical representation of the function integrates these various elements, offering a coherent visual narrative of how the function behaves across its domain. The graph confirms the analytical findings and provides an intuitive understanding of the function’s dynamics.

Thus, by systematically considering its end behavior, vertical asymptotes, intercepts, symmetry, and graphical representation, we can develop a comprehensive understanding of the rational function f(x) = (2x^4 + 9) / (3x^2 - 7). This holistic approach not only clarifies the specific characteristics of this function but also illustrates general principles applicable to analyzing other rational functions.