Analyzing The Rational Function F(x)

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Hey math enthusiasts! Today, we're diving deep into the fascinating world of rational functions. Specifically, we'll be breaking down and analyzing the function f(x) = (x^2 + 11) / (6x^2 - 5x - 4). Let's get started and unravel all its secrets! This function, a ratio of two polynomials, offers a rich landscape for exploration, revealing important concepts like domain, intercepts, asymptotes, and behavior at infinity. Understanding these components is critical to sketch the graph and correctly interpret its behavior. Analyzing rational functions is a fundamental skill in calculus and other higher-level math courses, so stick around because this is going to be useful!

Unveiling the Domain of f(x): Where Does it Live?

First things first, let's nail down the domain of our function. The domain essentially describes all the possible x-values we can plug into the function without causing any mathematical mayhem. With rational functions, the potential trouble spots are values of x that make the denominator equal to zero. Why? Because dividing by zero is a big no-no! So, to find the domain, we need to find the values of x that make 6x^2 - 5x - 4 = 0.

To find these values, we can use the quadratic formula. For a quadratic equation in the form ax^2 + bx + c = 0, the quadratic formula is: x = (-b ± √(b^2 - 4ac)) / 2a. In our case, a = 6, b = -5, and c = -4. Plugging these values into the formula, we get:

x = (5 ± √((-5)^2 - 4 * 6 * -4)) / (2 * 6) x = (5 ± √(25 + 96)) / 12 x = (5 ± √121) / 12 x = (5 ± 11) / 12

So, we have two possible solutions:

x = (5 + 11) / 12 = 16 / 12 = 4/3 x = (5 - 11) / 12 = -6 / 12 = -1/2

Therefore, the domain of f(x) is all real numbers except x = 4/3 and x = -1/2. In interval notation, the domain is (-∞, -1/2) ∪ (-1/2, 4/3) ∪ (4/3, ∞). This means the graph of the function won't exist at these x-values; these are where we'll find vertical asymptotes, which we'll discuss later. Understanding the domain is a cornerstone of function analysis because it immediately tells us where our function is defined and where it's not. Keep this in mind when graphing because it helps avoid confusion and ensures the graph is correct!

Intercepting Success: Finding the x and y Intercepts

Next up, let's hunt for the intercepts. Intercepts are the points where the graph of the function crosses the x-axis (x-intercept) or the y-axis (y-intercept). These are super useful for sketching a graph, as they provide us with anchor points.

  • Finding the x-intercept(s): To find the x-intercepts, we set f(x) = 0 and solve for x. In our function, f(x) = (x^2 + 11) / (6x^2 - 5x - 4). So, we need to solve: 0 = (x^2 + 11) / (6x^2 - 5x - 4)

    A fraction is equal to zero only when its numerator is zero. Thus, we focus on x^2 + 11 = 0. This equation simplifies to x^2 = -11. Since the square of a real number can never be negative, there are no real solutions for x. This means our function has no x-intercepts.

  • Finding the y-intercept: To find the y-intercept, we set x = 0 and solve for f(x). So, we plug in x = 0 into our function: f(0) = (0^2 + 11) / (6 * 0^2 - 5 * 0 - 4) f(0) = 11 / -4 = -11/4.

    Therefore, the y-intercept is at the point (0, -11/4) or (0, -2.75). The y-intercept is a single point on the graph where the curve crosses the vertical axis. Finding it is usually quite easy as it only involves a simple substitution, giving you one point to start with your graph.

Asymptotes: Guiding Lines of the Function

Let's move on to asymptotes, the invisible lines that guide the function's behavior. Asymptotes are lines that the graph of a function approaches but never quite touches. There are three main types of asymptotes: vertical, horizontal, and oblique (slant).

  • Vertical Asymptotes: We've already touched on vertical asymptotes when we discussed the domain. Vertical asymptotes occur at the values of x where the denominator of the rational function is equal to zero (and the numerator is not zero at the same x value). We found these values when determining the domain, namely, x = 4/3 and x = -1/2. So, our function has vertical asymptotes at x = 4/3 and x = -1/2. This tells us that as x gets closer and closer to these values, the function's value either increases or decreases without bound. When drawing the graph, vertical asymptotes appear as vertical dashed lines.

  • Horizontal Asymptotes: To find the horizontal asymptote, we consider the behavior of the function as x approaches positive or negative infinity. This depends on the degrees of the numerator and denominator polynomials. In our case, the degree of the numerator (x^2 + 11) is 2, and the degree of the denominator (6x^2 - 5x - 4) is also 2. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. In our function, the leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 6. Therefore, the horizontal asymptote is y = 1/6. This means as x becomes very large (positive or negative), the value of f(x) gets closer and closer to 1/6.

  • Oblique (Slant) Asymptotes: Oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. Since the degrees of our numerator and denominator are equal, we do not have an oblique asymptote.

Understanding asymptotes is key because they essentially define the boundaries of the function's behavior. They reveal how the function behaves as x moves towards infinity or towards specific restricted values. In the graphing stage, these dashed lines play a vital role in outlining the shape and overall behavior of the rational function. Remember the types of asymptotes, how they are calculated, and how they help the graph!

Unveiling Function Behavior: Analyzing End Behavior

Let's delve deeper into the function's behavior. We've talked about the horizontal asymptote, but let's formalize how to describe the function's end behavior. This refers to what happens to the function's value as x approaches positive infinity (x → ∞) and as x approaches negative infinity (x → -∞).

We've already determined that our function has a horizontal asymptote at y = 1/6. This tells us a lot about the end behavior. As x approaches both positive and negative infinity, the function's value approaches 1/6. Formally, we can write this as:

  • As x → ∞, f(x) → 1/6
  • As x → -∞, f(x) → 1/6

This end behavior is defined by the horizontal asymptote. The function's graph will get increasingly close to the line y=1/6 on both the left and right sides, but it will never touch or cross it (depending on certain conditions that aren't present here). This characteristic end behavior is a direct result of the nature of rational functions, and it provides valuable insight into the overall shape and scope of the graph. This understanding is key in many practical applications because it shows the long-term trend of the function's output as the input becomes very large or very small.

Graphing and Analysis: Putting it All Together

Now, let's pull all our findings together to sketch a graph of f(x). We can then use this graph to analyze the function fully.

  1. Domain: (-∞, -1/2) ∪ (-1/2, 4/3) ∪ (4/3, ∞). This gives us the intervals where the function is defined. We know there are vertical asymptotes at x = -1/2 and x = 4/3.
  2. Intercepts: No x-intercepts. The y-intercept is at (0, -11/4).
  3. Asymptotes: Vertical asymptotes at x = -1/2 and x = 4/3. Horizontal asymptote at y = 1/6.

With this information, we can start by drawing the vertical and horizontal asymptotes as dashed lines. Then, we can plot the y-intercept (0, -11/4). We can also choose a few test points in each interval defined by the vertical asymptotes to see if the curve is above or below the horizontal asymptote. For instance, testing a point like x = -1 gives f(-1) = (1+11)/(6+5-4) = 12/7, so we know the graph is above the horizontal asymptote on the interval between -∞ and -1/2. Testing a point like x = 0 gives -11/4, which we already knew. Testing a point like x=1 gives f(1) = (1+11)/(6-5-4) = -12/3 = -4, so on the interval between x = -1/2 and x = 4/3 the graph is well below the x-axis. As x becomes large, we know it approaches the horizontal asymptote y=1/6. By connecting these points and considering the behavior around the asymptotes, we can sketch a reasonably accurate graph.

Conclusion: Mastering Rational Function Analysis

And there you have it, folks! We've meticulously analyzed the rational function f(x) = (x^2 + 11) / (6x^2 - 5x - 4). We've explored its domain, found its intercepts, identified its asymptotes, and described its end behavior. The ability to perform such an analysis is a critical skill for any math student, and it is a key component to understanding the behavior of complex functions. Remember, practice makes perfect! So, grab some more rational functions and start digging in. Happy calculating!