Comprehensive Analysis Of The Function F(x) = (2x + 3) / (3x + 1)
In the realm of mathematics, functions serve as fundamental building blocks for modeling and understanding relationships between variables. Among the vast array of functions, rational functions hold a special place due to their unique characteristics and applications. This article delves into the intricacies of the rational function f(x) = (2x + 3) / (3x + 1), exploring its domain, range, intercepts, asymptotes, and overall behavior. Through a comprehensive analysis, we aim to provide a thorough understanding of this function and its properties.
Understanding the Domain of f(x) = (2x + 3) / (3x + 1)
When analyzing any function, the domain, which represents the set of all possible input values (x-values) for which the function is defined, is a crucial aspect to consider. For rational functions like f(x) = (2x + 3) / (3x + 1), the domain is restricted by the denominator. Specifically, the denominator cannot be equal to zero, as division by zero is undefined in mathematics. To determine the domain, we need to identify the values of x that would make the denominator, 3x + 1, equal to zero. Setting 3x + 1 = 0 and solving for x, we get:
3x + 1 = 0
3x = -1
x = -1/3
This indicates that x = -1/3 is the only value that makes the denominator zero. Therefore, the domain of the function f(x) = (2x + 3) / (3x + 1) includes all real numbers except for x = -1/3. In interval notation, this can be expressed as:
Domain: (-∞, -1/3) ∪ (-1/3, ∞)
Understanding the domain is essential because it helps us identify the values for which the function is defined and where it might exhibit unique behavior, such as vertical asymptotes. In this case, the exclusion of x = -1/3 from the domain signifies a potential vertical asymptote at this point, which we will explore further in the subsequent sections.
Unveiling the Range of f(x) = (2x + 3) / (3x + 1)
The range of a function encompasses the set of all possible output values (y-values) that the function can produce. Determining the range of a rational function like f(x) = (2x + 3) / (3x + 1) requires a slightly more intricate approach compared to finding the domain. One effective method involves analyzing the function's behavior as x approaches positive and negative infinity, as well as considering any horizontal asymptotes.
To find the horizontal asymptote, we examine the degrees of the numerator and denominator polynomials. In this case, both the numerator (2x + 3) and the denominator (3x + 1) have a degree of 1. When the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients of the numerator and denominator. Here, the leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 3. Therefore, the horizontal asymptote is y = 2/3.
This horizontal asymptote suggests that as x approaches positive or negative infinity, the function's output will approach 2/3 but never actually reach it. However, to definitively determine the range, we need to consider whether the function can take on all values except for 2/3. To do this, we can set y = (2x + 3) / (3x + 1) and solve for x:
y = (2x + 3) / (3x + 1)
y(3x + 1) = 2x + 3
3xy + y = 2x + 3
3xy - 2x = 3 - y
x(3y - 2) = 3 - y
x = (3 - y) / (3y - 2)
This equation expresses x in terms of y. For x to be a real number, the denominator (3y - 2) cannot be zero. Solving 3y - 2 = 0, we find y = 2/3. This confirms that the function cannot take on the value 2/3. Therefore, the range of the function f(x) = (2x + 3) / (3x + 1) includes all real numbers except for y = 2/3. In interval notation, the range is:
Range: (-∞, 2/3) ∪ (2/3, ∞)
Understanding the range provides valuable insights into the function's output behavior and helps us identify any limitations on the values it can produce. In this case, the exclusion of y = 2/3 from the range is directly linked to the horizontal asymptote of the function.
Locating Intercepts: Where f(x) Meets the Axes
Intercepts are the points where a function's graph intersects the coordinate axes. They provide valuable information about the function's behavior and can aid in sketching its graph. There are two types of intercepts to consider: x-intercepts and y-intercepts.
X-Intercepts
X-intercepts are the points where the graph intersects the x-axis. At these points, the y-value (or f(x) value) is equal to zero. To find the x-intercepts of f(x) = (2x + 3) / (3x + 1), we set f(x) = 0 and solve for x:
0 = (2x + 3) / (3x + 1)
For a fraction to be equal to zero, the numerator must be zero. Therefore, we set 2x + 3 = 0 and solve for x:
2x + 3 = 0
2x = -3
x = -3/2
This indicates that the function has one x-intercept at the point (-3/2, 0).
Y-Intercepts
Y-intercepts are the points where the graph intersects the y-axis. At these points, the x-value is equal to zero. To find the y-intercept of f(x) = (2x + 3) / (3x + 1), we substitute x = 0 into the function:
f(0) = (2(0) + 3) / (3(0) + 1)
f(0) = 3 / 1
f(0) = 3
This indicates that the function has one y-intercept at the point (0, 3).
By identifying the intercepts, we gain crucial anchor points for sketching the graph of the function. These points, along with the asymptotes, provide a framework for understanding the function's overall shape and behavior.
Asymptotes: Guiding the Behavior of f(x)
Asymptotes are lines that a function's graph approaches but never actually touches or crosses. They serve as guidelines for the function's behavior as x approaches infinity or certain specific values. Rational functions, like f(x) = (2x + 3) / (3x + 1), can have two types of asymptotes: vertical asymptotes and horizontal asymptotes.
Vertical Asymptotes
Vertical asymptotes occur at values of x where the denominator of the rational function is equal to zero, but the numerator is not zero. As we determined earlier, the denominator of f(x) = (2x + 3) / (3x + 1) is zero when x = -1/3. To confirm that this is a vertical asymptote, we need to ensure that the numerator is not also zero at this point. Substituting x = -1/3 into the numerator, we get:
2(-1/3) + 3 = -2/3 + 3 = 7/3
Since the numerator is not zero when x = -1/3, we can conclude that there is a vertical asymptote at x = -1/3. This means that the graph of the function will approach the vertical line x = -1/3 but never actually cross it.
Horizontal Asymptotes
Horizontal asymptotes describe the function's behavior as x approaches positive or negative infinity. To find the horizontal asymptote, we compare the degrees of the numerator and denominator polynomials. As we discussed earlier, both the numerator (2x + 3) and the denominator (3x + 1) have a degree of 1. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients, which is y = 2/3.
This indicates that as x becomes very large (positive or negative), the function's output will approach the horizontal line y = 2/3 but never actually reach it.
By identifying the asymptotes, we gain a deeper understanding of the function's long-term behavior and how it behaves near specific points. These asymptotes, along with the intercepts, provide a comprehensive framework for sketching the graph of the function.
Graphing f(x) = (2x + 3) / (3x + 1): A Visual Representation
To create an accurate graph of f(x) = (2x + 3) / (3x + 1), we can utilize the information we've gathered about its domain, range, intercepts, and asymptotes. Here's a step-by-step approach:
- Draw the Asymptotes: Start by drawing the vertical asymptote at x = -1/3 and the horizontal asymptote at y = 2/3. These lines will serve as guides for the graph.
- Plot the Intercepts: Plot the x-intercept at (-3/2, 0) and the y-intercept at (0, 3). These points provide anchor points for the graph.
- Choose Test Points: Select additional x-values in the intervals defined by the vertical asymptote and intercepts. For example, we could choose x = -2, x = -1, x = -1/6 and x = 1. Evaluate the function at these points to determine the corresponding y-values.
- Sketch the Graph: Using the asymptotes as guides and the plotted points as anchors, sketch the graph of the function. Remember that the graph will approach the asymptotes but never cross them (except possibly the horizontal asymptote in the middle of the graph).
By following these steps, you can create a visual representation of the function f(x) = (2x + 3) / (3x + 1) that accurately reflects its behavior and properties. The graph provides a powerful tool for understanding the function's characteristics and how it relates to the domain, range, intercepts, and asymptotes.
Conclusion
In conclusion, the function f(x) = (2x + 3) / (3x + 1) exemplifies the characteristics and behaviors of rational functions. Through a detailed analysis, we have explored its domain, which excludes x = -1/3, and its range, which excludes y = 2/3. We identified the x-intercept at (-3/2, 0) and the y-intercept at (0, 3), providing key points for graphing the function. Furthermore, we determined the vertical asymptote at x = -1/3 and the horizontal asymptote at y = 2/3, which guide the function's behavior as x approaches specific values and infinity. By combining these insights, we can accurately sketch the graph of the function and gain a comprehensive understanding of its properties and behavior. This exploration highlights the importance of analyzing various aspects of a function to fully grasp its mathematical nature and applications. Understanding these concepts is crucial for further studies in mathematics and related fields. The ability to analyze rational functions like this one opens doors to solving more complex problems and understanding real-world phenomena modeled by such functions. This thorough examination of f(x) = (2x + 3) / (3x + 1) serves as a strong foundation for future mathematical endeavors and reinforces the significance of function analysis in mathematics.
