Understanding Function Values Of F(x) = 1/(x+1) Near X = -1

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Hey guys! Let's dive into a super interesting math problem today. We're going to explore the function F(x) = 1/(x+1) and figure out what values it can take when x gets really, really close to -1. This is a classic example of understanding limits and how functions behave near certain points. So, buckle up, and let's get started!

Understanding the Function F(x) = 1/(x+1)

First, let's break down what this function actually means. The function F(x) = 1/(x+1) is a rational function. Rational functions are basically fractions where both the numerator (the top part) and the denominator (the bottom part) are polynomials. In our case, the numerator is simply 1, and the denominator is x + 1. The key thing to notice here is what happens when x gets close to -1. If x is exactly -1, the denominator becomes -1 + 1 = 0. And, as we all know, dividing by zero is a big no-no in the math world – it's undefined!

This means that the function has a vertical asymptote at x = -1. A vertical asymptote is like an invisible barrier that the function gets closer and closer to but never actually touches. To truly understand how the function behaves, we have to think about what happens as x approaches -1 from both sides – from values slightly smaller than -1 and from values slightly larger than -1.

Approaching -1 from the Left (Values Less Than -1)

Let's think about what happens when x is a little bit less than -1. For instance, what if x = -1.01? Then, x + 1 would be -1.01 + 1 = -0.01. So, F(x) would be 1/(-0.01) = -100. See what's happening? As x gets closer to -1 from the left, the denominator (x + 1) becomes a very small negative number, and the function value F(x) becomes a large negative number. If we took an even smaller value, like x = -1.0001, then x + 1 = -0.0001, and F(x) = 1/(-0.0001) = -10,000. The closer x gets to -1 from the left, the more negative F(x) becomes – it heads towards negative infinity! This is a crucial concept to grasp.

Approaching -1 from the Right (Values Greater Than -1)

Now, let's consider what happens when x is a little bit greater than -1. Suppose x = -0.99. Then, x + 1 = -0.99 + 1 = 0.01. So, F(x) = 1/(0.01) = 100. Notice the difference? As x approaches -1 from the right, the denominator (x + 1) becomes a very small positive number, and the function value F(x) becomes a large positive number. If x = -0.9999, then x + 1 = 0.0001, and F(x) = 1/(0.0001) = 10,000. So, the closer x gets to -1 from the right, the more positive F(x) becomes – it heads towards positive infinity! This contrasting behavior from the left and right sides is really important for understanding the function's overall behavior near x = -1.

Analyzing the Answer Choices

Okay, now that we have a solid understanding of how the function F(x) behaves near x = -1, let's take a look at the answer choices and see which one makes the most sense.

We have four options:

  • A. 0.01
  • B. -0.01
  • C. -1
  • D. -10,000

We know that as x gets closer to -1, F(x) becomes a very large positive or a very large negative number. Options A, B, and C (0.01, -0.01, and -1) are all relatively small numbers. They just don't fit the pattern we've observed. Option D, on the other hand, is -10,000. This is a large negative number, which perfectly aligns with what we found when x approaches -1 from the left. Remember, when x is slightly less than -1, F(x) becomes a very large negative number. So, -10,000 is a very plausible value for F(x) when x is close to -1.

Choosing the Correct Answer

Based on our analysis, the answer choice that could be a value of F(x) when x is close to -1 is D. -10,000. The other options are simply too small to reflect the function's behavior as it approaches its vertical asymptote.

Key Takeaways

  • The function F(x) = 1/(x+1) has a vertical asymptote at x = -1.
  • As x approaches -1 from the left, F(x) approaches negative infinity.
  • As x approaches -1 from the right, F(x) approaches positive infinity.
  • Large positive and negative numbers are plausible values for F(x) when x is close to -1.

Exploring Further

This problem is a great introduction to the concept of limits in calculus. Understanding how functions behave near points where they are undefined is fundamental to many advanced mathematical concepts. If you found this interesting, I encourage you to explore limits and asymptotes further. You can try graphing the function F(x) = 1/(x+1) to visualize its behavior near x = -1. You can also try working through similar problems with different rational functions to solidify your understanding.

Conclusion

So, there you have it! We've successfully navigated the function F(x) = 1/(x+1) and figured out which value could be a result when x is close to -1. Remember, math isn't just about getting the right answer; it's about understanding why the answer is correct. By breaking down the problem, thinking about the behavior of the function, and analyzing the answer choices, we were able to arrive at the correct solution. Keep exploring, keep questioning, and most importantly, keep having fun with math!