Domain And Range Of A Rational Function Explained

In mathematics, understanding the domain and range of a function is crucial for analyzing its behavior and properties. The domain represents all possible input values (often x-values) for which the function is defined, while the range represents all possible output values (often y-values) that the function can produce. When dealing with rational functions, which are functions expressed as a ratio of two polynomials, determining the domain and range requires careful consideration of potential restrictions, such as division by zero and other mathematical impossibilities.

In this article, we will delve into a step-by-step approach to finding the domain and range of a complex rational function. Specifically, we will analyze the function:

$\qquad f(x) = \frac{4(x+2)(x+1)(3x+4)(2x-3)}{3(x-2)(x^2-1)(2x-3)(x+2)^4}$

This function presents several challenges due to its intricate structure, including multiple factors in both the numerator and the denominator. By systematically identifying and addressing these challenges, we can accurately determine the function's domain and range. Let’s embark on this mathematical journey and unravel the intricacies of this rational function.

Determining the Domain of the Function

To find the domain of the function,

$\qquad f(x) = \frac{4(x+2)(x+1)(3x+4)(2x-3)}{3(x-2)(x^2-1)(2x-3)(x+2)^4}$

we need to identify all values of x for which the function is defined. A rational function is undefined when its denominator is equal to zero. Therefore, our first step is to find the values of x that make the denominator zero.

The denominator of the function is:

$\qquad 3(x-2)(x^2-1)(2x-3)(x+2)^4$

To find the zeroes, we set each factor equal to zero and solve for x:

1. $\qquad x - 2 = 0 \implies x = 2$
2. $\qquad x^2 - 1 = 0 \implies x^2 = 1 \implies x = \pm 1$
3. $\qquad 2x - 3 = 0 \implies 2x = 3 \implies x = \frac{3}{2}$
4. $\qquad (x + 2)^4 = 0 \implies x + 2 = 0 \implies x = -2$

Thus, the values of x that make the denominator zero are x = -2, x = -1, x = 1, x = 3/2, and x = 2. These values must be excluded from the domain.

Therefore, the domain of the function f(x) is all real numbers except x = -2, x = -1, x = 1, x = 3/2, and x = 2. In interval notation, the domain can be expressed as:

$\qquad D: (-\infty, -2) \cup (-2, -1) \cup (-1, 1) \cup (1, \frac{3}{2}) \cup (\frac{3}{2}, 2) \cup (2, \infty)$

This notation indicates that the domain includes all real numbers less than -2, between -2 and -1, between -1 and 1, between 1 and 3/2, between 3/2 and 2, and greater than 2. Understanding these exclusions is critical for accurately analyzing the behavior of the function.

Simplifying the Function

Before determining the range, it is often helpful to simplify the function. Simplification can reveal hidden structures and make the function easier to analyze. Our original function is:

$\qquad f(x) = \frac{4(x+2)(x+1)(3x+4)(2x-3)}{3(x-2)(x^2-1)(2x-3)(x+2)^4}$

We can simplify this function by canceling common factors in the numerator and the denominator. Notice that $(2x - 3)$ and one factor of $(x + 2)$ appear in both the numerator and the denominator. Also, $x^2 - 1$ can be factored as $(x - 1)(x + 1)$. Thus, we have:

$\qquad f(x) = \frac{4(x+2)(x+1)(3x+4)(2x-3)}{3(x-2)(x-1)(x+1)(2x-3)(x+2)^4}$

Canceling the common factors, we get:

$\qquad f(x) = \frac{4(3x+4)}{3(x-2)(x-1)(x+2)^3}$

However, it’s crucial to remember that even though we have simplified the function, the original domain restrictions still apply. The simplified function does not explicitly show the excluded values x = -1 and x = 3/2, but they remain points where the original function is undefined. The simplified form helps in analyzing the function's behavior but does not change its fundamental restrictions.

Determining the Range of the Function

Finding the range of a rational function can be more challenging than finding the domain. The range consists of all possible output values of the function. For complex rational functions, there is no straightforward method to determine the range algebraically. We typically rely on a combination of analytical techniques and graphical analysis.

One approach is to consider the behavior of the function as x approaches the values excluded from the domain and as x approaches positive and negative infinity. These limits can provide insights into the possible y-values the function can take. Additionally, identifying any horizontal or vertical asymptotes can help define the boundaries of the range.

However, for the given function:

$\qquad f(x) = \frac{4(3x+4)}{3(x-2)(x-1)(x+2)^3}$

determining the range algebraically is quite complex due to the presence of the cubic term in the denominator. Instead, we can use graphical analysis or numerical methods to approximate the range. Using a graphing calculator or software, we can plot the function and observe its behavior.

The graph of the function reveals that it has vertical asymptotes at x = -2, x = 1, and x = 2, as we determined earlier. It also shows that the function approaches 0 as x approaches positive or negative infinity, indicating a horizontal asymptote at y = 0. Furthermore, there are local maxima and minima that affect the range.

Without precise algebraic methods, we can approximate the range by observing the graph. The function takes on values across the real number line, but there might be intervals where the function does not exist or has gaps due to asymptotes. The accurate determination of these intervals often requires advanced calculus techniques or numerical computation.

For this specific function, providing an exact algebraic expression for the range is beyond the scope of this basic analysis. Typically, the range is determined through advanced calculus methods, such as finding critical points and analyzing concavity, or through numerical and graphical approximations.

Possible Answers and Discussion

The original question provides two possible answers for the domain and range:

• A. $D:(-\infty,-1) \cup(-1,3 / 2) \cup(3 / 2, \infty) R:(-\infty, 4) \cup(4, \infty$
• B. $D:(-\infty,-1)$

Based on our analysis, the correct domain should exclude -2, -1, 1, 3/2, and 2. Option A correctly excludes -1 and 3/2 but does not account for -2, 1, and 2. Therefore, Option A is incomplete.

The correct domain, as we derived, is:

$\qquad D: (-\infty, -2) \cup (-2, -1) \cup (-1, 1) \cup (1, \frac{3}{2}) \cup (\frac{3}{2}, 2) \cup (2, \infty)$

As for the range, Option A suggests $R:(-\infty, 4) \cup(4, \infty)$, which is a simplification and likely an approximation. The actual range is more complex and would require further analysis beyond basic algebraic methods.

In conclusion, finding the domain of the function involves identifying and excluding values that make the denominator zero. Determining the range, however, is more intricate and often requires graphical or numerical methods, or advanced calculus techniques. This detailed analysis ensures a comprehensive understanding of the function's behavior and characteristics.

Conclusion

In this comprehensive exploration, we have meticulously navigated the process of determining the domain and range of a complex rational function. Our journey began with identifying the critical values that exclude certain inputs from the domain, ensuring that the function remains well-defined. We systematically pinpointed these values by setting the denominator of the function to zero and solving for x. This step is fundamental in understanding the function's boundaries and potential discontinuities.

Next, we ventured into the simplification of the function, a crucial step that not only unveils hidden structures but also streamlines subsequent analysis. By canceling out common factors in the numerator and the denominator, we transformed the function into a more manageable form. However, we emphasized the importance of retaining the original domain restrictions, as the simplified form should not mask the inherent limitations of the function.

Determining the range presented a more formidable challenge, underscoring the complexity involved in characterizing all possible output values of a rational function. While an algebraic determination proved intricate, we advocated for a multi-faceted approach that incorporates graphical analysis and numerical methods. Visualizing the function's behavior through graphs offers invaluable insights into its range, revealing patterns, asymptotes, and potential extrema.

Throughout this discussion, we've underscored the significance of understanding the interplay between algebraic manipulation and graphical interpretation in analyzing functions. By mastering these techniques, one can gain a profound understanding of a function's behavior, including its domain, range, and other essential properties.

In summary, finding the domain and range of a rational function is a methodical yet rewarding process. It requires a keen eye for detail, a solid grasp of algebraic principles, and the willingness to leverage graphical tools. As we've demonstrated, a systematic approach not only yields accurate results but also cultivates a deeper appreciation for the elegance and intricacies of mathematical functions.