Removable Discontinuity: F(x) = (5x+10)/(x^2+7x+10)
Hey guys! Today, we're diving into the fascinating world of removable discontinuities. Specifically, we're going to figure out where the removable discontinuity is located in the function f(x) = (5x + 10) / (x^2 + 7x + 10). Don't worry, it's not as scary as it looks! We'll break it down step-by-step so you can confidently tackle these kinds of problems.
Understanding Removable Discontinuities
Before we jump into the function itself, let's quickly recap what a removable discontinuity actually is. Imagine a graph with a tiny hole in it – that's essentially what we're talking about. A removable discontinuity, also known as a hole, occurs at a point where the function is undefined, but could be defined to make the function continuous at that point. This usually happens when a factor in the numerator and denominator of a rational function cancels out. This cancellation creates a 'hole' rather than an asymptote because the factor doesn't cause the function to approach infinity.
Removable discontinuities are super important in calculus and analysis. They show us where a function almost behaves nicely, and understanding them is crucial for things like finding limits and derivatives. It’s not just about spotting the hole; it’s about understanding why it’s there and what it tells us about the function's behavior. In the context of real-world applications, these discontinuities can represent situations where a model breaks down at a specific point but is otherwise well-behaved. For example, in physics, it might represent a point where a certain approximation is no longer valid.
Identifying and dealing with removable discontinuities is also vital for various mathematical operations. When you're integrating a function, for instance, you need to be aware of these discontinuities because they can affect the result. Similarly, when analyzing the stability of a system modeled by a function, the presence and nature of discontinuities can provide critical insights. Remember, guys, that a solid grasp of these concepts can make a significant difference in your problem-solving abilities.
Step 1: Factor the Numerator and Denominator
The first step in finding removable discontinuities is to factor both the numerator and the denominator of our function. This will help us identify any common factors that can be canceled out.
Our function is f(x) = (5x + 10) / (x^2 + 7x + 10).
Let's start with the numerator: 5x + 10. We can factor out a 5:
5x + 10 = 5(x + 2)
Now, let's factor the denominator: x^2 + 7x + 10. We're looking for two numbers that multiply to 10 and add up to 7. Those numbers are 2 and 5. So, we can factor the denominator as:
x^2 + 7x + 10 = (x + 2)(x + 5)
Now, our function looks like this:
f(x) = 5(x + 2) / (x + 2)(x + 5)
Factoring is like the detective work of mathematics. It's about breaking things down to their simplest components to reveal hidden relationships. In this case, factoring the numerator and denominator allows us to see if there are any common factors lurking. Without this step, identifying removable discontinuities would be much more challenging. Factoring not only simplifies the function but also makes it easier to analyze its behavior, particularly around points where the denominator might be zero.
Moreover, this step is a fundamental skill that extends far beyond just finding removable discontinuities. It’s used in solving equations, simplifying expressions, and understanding the structure of polynomials and rational functions. Mastering factoring techniques is crucial for success in algebra and calculus. Think of it as learning the alphabet of the mathematical language – once you know the letters, you can form words, sentences, and even write entire stories. So, guys, make sure you're comfortable with factoring, as it's a tool you'll be using throughout your mathematical journey.
Step 2: Identify Common Factors and Cancel Them
Alright, now that we've factored the numerator and denominator, we can spot any common factors. In our factored function,
f(x) = 5(x + 2) / (x + 2)(x + 5),
we see that (x + 2) appears in both the numerator and the denominator. This is our key to finding the removable discontinuity!
We can cancel out the (x + 2) terms, which simplifies our function to:
f(x) = 5 / (x + 5), where x ≠-2
The crucial part here is noting that x ≠-2. Even though we've canceled out the (x + 2) term, the original function was undefined at x = -2 because it would have resulted in division by zero. This is what creates the “hole” or removable discontinuity.
Think of canceling common factors as simplifying a fraction. Just like reducing 6/8 to 3/4 doesn't change its value, canceling (x + 2) doesn't change the function's value except at the point where the canceled factor equals zero. This exception is precisely where the removable discontinuity occurs. It's like removing a single brick from a wall – the wall is still mostly intact, but there's a gap where the brick used to be.
This step is also a great illustration of how mathematical operations can have subtle yet significant consequences. We've simplified the function, but we've also revealed something important about its nature. The act of canceling the common factor highlights the point where the function is undefined, giving us crucial information about its behavior. So, guys, pay attention to these subtle details – they often hold the key to solving the puzzle.
Step 3: Determine the Location of the Removable Discontinuity
We've simplified our function to f(x) = 5 / (x + 5), but we haven't forgotten that x ≠-2 due to the cancellation of the (x + 2) factor. This is where our removable discontinuity lives!
The removable discontinuity occurs at the x-value that makes the canceled factor equal to zero. In this case, we canceled (x + 2), so we set (x + 2) = 0 and solve for x:
x + 2 = 0 x = -2
So, the removable discontinuity is located at x = -2. To find the y-coordinate of the discontinuity (the “hole” in the graph), we plug x = -2 into the simplified function f(x) = 5 / (x + 5):
f(-2) = 5 / (-2 + 5) = 5 / 3
Therefore, the removable discontinuity is located at the point (-2, 5/3).
Finding the exact location of the discontinuity is like pinpointing a specific address on a map. We know there's a break in the function, but we want to know precisely where it is. This is important because it gives us a complete picture of the function's behavior. We now know that the function behaves like 5 / (x + 5) everywhere except at x = -2, where there's a tiny gap.
This process also highlights the importance of paying attention to detail. We couldn't just look at the simplified function to find the discontinuity; we had to remember the factor that we canceled. It's a reminder that mathematical operations can sometimes hide information, and it's our job to uncover it. Guys, this attention to detail is what separates a good problem-solver from a great one. So, always double-check your work and make sure you haven't overlooked anything important.
Step 4: Visualize the Discontinuity (Optional)
While we've found the location of the removable discontinuity algebraically, it can be helpful to visualize it graphically. You can use graphing software or a calculator to plot the function f(x) = (5x + 10) / (x^2 + 7x + 10) or its simplified form f(x) = 5 / (x + 5). You'll notice that the graph looks like the graph of 5 / (x + 5), but there's a tiny hole at the point (-2, 5/3).
Visualizing the discontinuity can solidify your understanding of what it actually means. It's one thing to calculate the coordinates of the hole, but it's another to see it on a graph. The visual representation makes the abstract concept of a discontinuity much more concrete. It's like seeing a blueprint of a building – you get a much better sense of the overall structure than just reading a description.
Moreover, graphing the function can help you catch any mistakes you might have made in your algebraic calculations. If your graph doesn't show a hole at the expected location, it's a sign that you need to go back and check your work. Graphing is a powerful tool for verifying your results and building your intuition about functions and their behavior. So, guys, don't underestimate the power of visualization – it can be a game-changer in your mathematical journey.
Conclusion
So, there you have it! We've successfully found the removable discontinuity of the function f(x) = (5x + 10) / (x^2 + 7x + 10), which is located at (-2, 5/3). Remember, the key steps are:
- Factor the numerator and denominator.
- Identify and cancel common factors.
- Determine the x-value that makes the canceled factor equal to zero.
- Plug that x-value into the simplified function to find the y-coordinate.
Understanding removable discontinuities is a valuable skill in calculus and beyond. It helps us analyze the behavior of functions and solve a wide range of problems. Keep practicing, and you'll become a pro at spotting and dealing with these “holes” in no time!
Finding removable discontinuities is more than just a mathematical exercise; it's a journey into the nuances of function behavior. It teaches us to look beyond the surface and understand the underlying structure. By mastering these concepts, guys, you're not just learning math – you're developing critical thinking skills that will serve you well in any field. So, keep exploring, keep questioning, and keep pushing your mathematical boundaries. You've got this!
