Analyzing The Function F(x) = (3x-1)/(2x-5): A Deep Dive
Hey guys! Today, we're going to dive deep into the analysis of a specific function: f(x) = (3x - 1) / (2x - 5). This type of function, a rational function, is super common in mathematics, and understanding its properties is crucial for calculus, pre-calculus, and even some areas of algebra. We’ll explore everything from its domain and range to its asymptotes and intercepts. So, buckle up, grab your pencils, and let's get started!
Understanding the Basics of Rational Functions
Before we jump into the specifics of f(x) = (3x - 1) / (2x - 5), let’s quickly recap what rational functions are all about. A rational function is essentially a function that can be expressed as the quotient of two polynomials. In simpler terms, it's a fraction where both the numerator and the denominator are polynomials. Our function fits this description perfectly, with (3x - 1) and (2x - 5) being linear polynomials. These functions often exhibit unique behaviors, especially around points where the denominator equals zero, leading to asymptotes and other interesting features. Recognizing these features is key to sketching their graphs and understanding their behavior. The general form of a rational function helps us identify potential vertical asymptotes, horizontal asymptotes, and other key characteristics. By understanding the underlying principles of rational functions, we'll be better equipped to analyze the specific function at hand and appreciate its nuances. So, keep this foundation in mind as we delve deeper into the analysis.
Finding the Domain: Where Does the Function Exist?
First things first, let's talk about the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions like ours, the main concern is the denominator. We need to figure out where the denominator, (2x - 5), equals zero because division by zero is a big no-no in math. To find these values, we set (2x - 5) = 0 and solve for x. This gives us x = 5/2. Therefore, the function is undefined at x = 5/2. So, the domain of f(x) is all real numbers except 5/2. We can write this in interval notation as (-∞, 5/2) ∪ (5/2, ∞). This means that the function can accept any x-value as input, except for x = 5/2. This understanding of the domain is crucial because it tells us where the function is valid and where we might encounter some interesting behavior, like a vertical asymptote, which we'll discuss later. Remember, identifying the domain is a foundational step in analyzing any function, as it sets the stage for understanding its overall behavior and limitations.
Identifying Asymptotes: Guiding the Graph
Now, let's explore asymptotes. Asymptotes are like invisible guide rails for the graph of a function. They're lines that the graph approaches but never quite touches. Rational functions often have two types of asymptotes: vertical and horizontal.
Vertical Asymptotes
Vertical asymptotes occur where the denominator of the function equals zero, which we already found to be x = 5/2. This is because as x gets closer and closer to 5/2, the denominator approaches zero, causing the function value to shoot off towards positive or negative infinity. On the graph, you'll see the function getting closer and closer to the vertical line x = 5/2 but never actually crossing it. Understanding vertical asymptotes is essential for sketching the graph of a rational function, as they dictate the behavior of the function near these critical points. They also highlight the values that are excluded from the domain, reinforcing the importance of domain analysis. So, the vertical asymptote at x = 5/2 is a key feature of our function.
Horizontal Asymptotes
Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To find the horizontal asymptote, we compare the degrees of the polynomials in the numerator and the denominator. In our case, both the numerator (3x - 1) and the denominator (2x - 5) have a degree of 1 (they're both linear). When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 2. So, the horizontal asymptote is y = 3/2. This means that as x gets extremely large (positive or negative), the function values will get closer and closer to 3/2. The horizontal asymptote provides valuable information about the long-term behavior of the function, indicating where the function tends to level off. It's another crucial element in understanding and sketching the graph of rational functions.
Finding Intercepts: Where the Function Crosses the Axes
Next up, let's find the intercepts. Intercepts are the points where the graph of the function crosses the x-axis and the y-axis. These points are super helpful for sketching the graph and understanding the function's behavior near the axes.
Y-intercept
To find the y-intercept, we set x = 0 and evaluate the function: f(0) = (3(0) - 1) / (2(0) - 5) = -1 / -5 = 1/5. So, the y-intercept is at the point (0, 1/5). This tells us where the function intersects the vertical axis, providing a starting point for sketching the graph. The y-intercept is always a straightforward calculation, making it a valuable piece of information in our analysis.
X-intercept
To find the x-intercept, we set f(x) = 0 and solve for x. This means we need to find where the numerator, (3x - 1), equals zero. Setting (3x - 1) = 0, we get x = 1/3. So, the x-intercept is at the point (1/3, 0). This indicates where the function crosses the horizontal axis, further defining the behavior of the graph. Finding the x-intercept is crucial because it helps us identify the roots or zeros of the function, which are fundamental in many mathematical applications.
Range of the Function: What Values Does it Output?
Now, let's tackle the range. The range of a function is the set of all possible output values (y-values) that the function can produce. For our rational function, finding the range involves considering the horizontal asymptote and the overall behavior of the function. Since the horizontal asymptote is y = 3/2, we know that the function will approach this value as x goes to infinity or negative infinity. However, we need to determine if the function actually takes on the value 3/2. To do this, we set f(x) = 3/2 and try to solve for x:
(3x - 1) / (2x - 5) = 3/2
Cross-multiplying, we get:
2(3x - 1) = 3(2x - 5) 6x - 2 = 6x - 15 -2 = -15
This equation has no solution, which means that the function never actually equals 3/2. Therefore, the range of f(x) is all real numbers except 3/2. We can write this in interval notation as (-∞, 3/2) ∪ (3/2, ∞). Determining the range is vital for a complete understanding of the function, as it tells us the full spectrum of possible output values. This, combined with the domain, provides a comprehensive picture of the function's behavior.
Sketching the Graph: Putting it All Together
Alright, guys, let's bring everything together and sketch the graph of f(x) = (3x - 1) / (2x - 5). We've gathered a bunch of crucial information:
- Domain: (-∞, 5/2) ∪ (5/2, ∞)
- Vertical Asymptote: x = 5/2
- Horizontal Asymptote: y = 3/2
- Y-intercept: (0, 1/5)
- X-intercept: (1/3, 0)
- Range: (-∞, 3/2) ∪ (3/2, ∞)
To sketch the graph, start by drawing the asymptotes as dashed lines. These will act as guides for the graph. Then, plot the intercepts. Now, think about the behavior of the function near the asymptotes and intercepts. As x approaches 5/2 from the left, the function will go towards negative infinity. As x approaches 5/2 from the right, the function will go towards positive infinity. As x goes to positive or negative infinity, the function will approach the horizontal asymptote y = 3/2. Using this information, you can sketch the two branches of the hyperbola. One branch will pass through the y-intercept and approach the asymptotes in the second and third quadrants. The other branch will pass through the x-intercept and approach the asymptotes in the first and fourth quadrants. Sketching the graph is a fantastic way to visualize all the analytical information we've gathered, making it easier to understand the function's overall behavior and characteristics.
Conclusion: Mastering Rational Function Analysis
So, guys, we've taken a comprehensive journey through the analysis of the rational function f(x) = (3x - 1) / (2x - 5). We’ve covered finding the domain, identifying asymptotes, determining intercepts, understanding the range, and finally, sketching the graph. By breaking down the function step-by-step, we've gained a solid understanding of its behavior and characteristics. This process is applicable to many other rational functions, making these skills incredibly valuable in mathematics. Remember, practice makes perfect! The more you analyze these types of functions, the easier it will become. Keep exploring, keep questioning, and keep learning! You've got this!