Analyzing F(x) = (x-7)/(x^2-9x+14): A Detailed Guide

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Analyzing f(x) = (x-7)/(x^2-9x+14): A Detailed Guide

Hey guys! Today, we're diving deep into the function f(x) = (x-7)/(x^2 - 9x + 14). This might look intimidating at first, but don't worry! We'll break it down step by step so you can understand everything about it. We'll cover simplifying the function, finding its domain, identifying any asymptotes, and sketching its graph. So, grab your thinking caps, and let's get started!

Simplifying the Function

First things first, let's simplify our function. Simplifying a function makes it easier to analyze and understand its behavior. Our function is f(x) = (x-7)/(x^2 - 9x + 14). Notice anything interesting about the denominator? That's right, it's a quadratic expression that can be factored! The denominator x^2 - 9x + 14 can be factored into (x - 7)(x - 2). So, we can rewrite our function as:

f(x) = (x - 7) / ((x - 7)(x - 2)).

Now, do you see it? The term (x - 7) appears in both the numerator and the denominator. This means we can cancel it out! However, it's super important to remember that we can only cancel out terms if x is not equal to 7. Because if x were 7, we'd be dividing by zero, which is a big no-no in mathematics! So, after canceling out the (x - 7) terms, we get:

f(x) = 1 / (x - 2), where x ≠ 7.

This simplified form is much easier to work with. It tells us that the function behaves like 1/(x - 2), except at x = 7. At x = 7, there's a hole in the graph, which we'll talk about later. This simplification highlights a crucial concept: always look for opportunities to simplify functions before analyzing them. It can save you a lot of headaches down the road and give you a clearer picture of what's going on. By factoring and canceling common terms, we've transformed a seemingly complex function into a much more manageable one.

Determining the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's all the x-values that you can plug into the function without causing any mathematical errors, like dividing by zero or taking the square root of a negative number. For our simplified function, f(x) = 1 / (x - 2), we need to identify any values of x that would make the denominator equal to zero. If the denominator is zero, the function is undefined at that point because division by zero is not allowed. So, let's find those values:

x - 2 = 0 implies x = 2.

This means that x = 2 is not in the domain of the function. Also, remember when we simplified the function? We had to make a note that x ≠ 7 because of the original function. So, x = 7 is also not in the domain. Therefore, the domain of the function f(x) is all real numbers except for x = 2 and x = 7. We can write this in interval notation as:

(-∞, 2) ∪ (2, 7) ∪ (7, ∞)

Understanding the domain is crucial because it tells us where the function is "valid." It helps us avoid making incorrect calculations or drawing inaccurate conclusions about the function's behavior. In this case, we know that the function is well-behaved for all real numbers except at x = 2 (where there's a vertical asymptote) and x = 7 (where there's a hole).

Identifying Asymptotes

Asymptotes are lines that a function approaches but never quite reaches. They give us important clues about the function's behavior as x approaches certain values or as x goes to infinity or negative infinity. There are mainly three types of asymptotes: vertical, horizontal, and oblique (or slant) asymptotes.

Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity (or negative infinity) as x approaches a certain value. These typically happen where the denominator of a simplified rational function equals zero. From our domain analysis, we know that x = 2 makes the denominator of the simplified function equal to zero. Therefore, there is a vertical asymptote at x = 2. This means that as x gets closer and closer to 2, the value of the function will either shoot up towards positive infinity or plummet down towards negative infinity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To find horizontal asymptotes, we need to examine the limit of the function as x goes to infinity and negative infinity. For our simplified function, f(x) = 1 / (x - 2), as x becomes very large (either positive or negative), the term 1 / (x - 2) approaches zero. Therefore, there is a horizontal asymptote at y = 0. This means that as you move further and further to the right or left on the graph, the function gets closer and closer to the x-axis but never actually touches it.

Oblique (Slant) Asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. In our simplified function, the degree of the numerator is 0 (since it's just a constant), and the degree of the denominator is 1. Therefore, there are no oblique asymptotes in this case. Understanding asymptotes helps us sketch the graph of the function accurately. They act as guidelines, showing us where the function is "allowed" to go and where it's "pushed away" from.

Identifying Holes

Remember when we simplified the function f(x) = (x - 7) / ((x - 7)(x - 2))? We canceled out the (x - 7) term, but we had to remember that x ≠ 7. This means that at x = 7, the original function is undefined, even though the simplified function is defined. This creates a "hole" in the graph at x = 7. To find the y-coordinate of the hole, we plug x = 7 into the simplified function:

f(7) = 1 / (7 - 2) = 1 / 5

So, there is a hole in the graph at the point (7, 1/5). A hole is a point where the function is undefined, but the graph "would have been" continuous if the function were defined there. It's like a tiny gap in the graph that you can't see unless you zoom in really close. Recognizing and identifying holes is important for accurately representing the function's behavior and understanding its true nature.

Sketching the Graph

Now that we've analyzed the function, found its domain, identified its asymptotes, and located the hole, we can finally sketch its graph! Here's how we can approach it:

  1. Draw the asymptotes: Draw the vertical asymptote at x = 2 and the horizontal asymptote at y = 0. These lines will guide your sketch.
  2. Plot the hole: Mark the hole at the point (7, 1/5). Remember to draw it as an open circle to indicate that the function is not defined at that point.
  3. Analyze the behavior around the vertical asymptote: As x approaches 2 from the left (i.e., values slightly less than 2), the function f(x) = 1 / (x - 2) approaches negative infinity. As x approaches 2 from the right (i.e., values slightly greater than 2), the function approaches positive infinity.
  4. Analyze the behavior as x approaches infinity: As x approaches positive infinity, the function approaches 0 from above (i.e., positive values). As x approaches negative infinity, the function approaches 0 from below (i.e., negative values).
  5. Plot a few additional points: Choose a few x-values in each region (to the left of the vertical asymptote, between the vertical asymptote and the hole, and to the right of the hole) and calculate the corresponding y-values. This will give you a better sense of the shape of the graph.
  6. Connect the dots: Draw smooth curves that follow the asymptotes, pass through the plotted points, and avoid the hole. Remember that the function is not defined at the vertical asymptote or at the hole.

By following these steps, you can create a reasonably accurate sketch of the graph of f(x) = (x - 7) / (x^2 - 9x + 14). The graph will show a curve that approaches the vertical asymptote at x = 2, approaches the horizontal asymptote at y = 0, and has a hole at the point (7, 1/5). Sketching the graph is a great way to visualize the function's behavior and confirm your analysis.

Conclusion

So, there you have it! We've successfully dissected the function f(x) = (x - 7) / (x^2 - 9x + 14). We simplified it, found its domain, identified its asymptotes and holes, and even sketched its graph. Remember, the key to understanding these functions is to break them down into smaller, manageable steps. Always look for opportunities to simplify, pay close attention to the domain, and use asymptotes as guides. With a little practice, you'll be analyzing rational functions like a pro! Keep exploring, keep learning, and most importantly, have fun with math! You got this!