Function Analysis: Domain, Limits, And Mathematical Proof

Oct 29, 2025 by Admin 58 views

Hey guys! Let's dive into some cool math stuff today. We're going to analyze a function, figure out its domain, calculate its limits, and even prove a mathematical statement. Buckle up, it's going to be a fun ride. The function we are looking at is defined as $f(x) = \frac{x^2}{x+1}$ . Let's break this down step by step to make sure we understand everything. We'll follow the instructions that we were given, and hopefully, we can get a good understanding of what's going on. This is going to be a fun exercise, so make sure to follow along and try it yourself. These types of exercises are super important, so try to grasp every bit of knowledge that you can. I am here to help you get through it all, so let's start.

Determining the Domain of the Function

Alright, first things first: determining the domain, or $D_f$ . The domain of a function is the set of all possible input values (x-values) for which the function is defined. When we look at our function, $f(x) = \frac{x^2}{x+1}$ , we see a fraction. Remember, in math, we can't divide by zero. So, we need to figure out what values of $x$ would make the denominator, $(x + 1)$ , equal to zero. This is one of the most important things to do, so make sure you get this down! It can be a little tricky sometimes, so be sure you take your time. This is where we pay attention to detail.

To find this out, we set the denominator equal to zero and solve for x:

$x + 1 = 0$

Subtract 1 from both sides:

$x = -1$

This means that when $x = -1$ , the denominator becomes zero, and the function is undefined. Therefore, the domain of the function, $D_f$ , is all real numbers except for -1. We can write this in a couple of ways:

$D_f = \{x \in \mathbb{R} \mid x \neq -1\}$
$D_f = (-\infty, -1) \cup (-1, +\infty)$

So, the domain is everything except for x = -1. This is a very important concept, so make sure to take your time and study it. This will help you in the long run, and it's a critical part of the process. I believe you can do it, so keep on working hard, and never give up. I am sure you can get it done! Keep up the good work, you're doing great!

Calculating the Limits of the Function

Next up, we need to calculate the limits of the function at the boundaries of its domain. Since our domain is all real numbers except for -1, we need to find the limits as $x$ approaches -1 and as $x$ approaches positive and negative infinity. This might seem complex at first, but don't worry, we'll break it down.

Limit as x approaches -1: We need to consider both sides of -1, that is, when $x$ approaches -1 from values less than -1 (from the left, denoted as $x \to -1^-$ ) and from values greater than -1 (from the right, denoted as $x \to -1^+$ ). As $x$ gets closer and closer to -1, the numerator $x^2$ gets closer to $(-1)^2 = 1$ . The denominator $x + 1$ approaches 0. If we approach -1 from the left, $x + 1$ approaches 0 from negative values, so the fraction goes to $-\infty$ . If we approach -1 from the right, $x + 1$ approaches 0 from positive values, so the fraction goes to $+\infty$ . Therefore, the limit does not exist. However, we can also see the limit of $f(x)$ when $x \to -1^-$ is $-\infty$ and when $x \to -1^+$ is $+\infty$ .
Limit as x approaches positive infinity: As $x$ gets infinitely large, the numerator $x^2$ grows much faster than the denominator $x + 1$ . We can rewrite the function by dividing both the numerator and denominator by $x$ : $f(x) = \frac{x}{1 + \frac{1}{x}}$ . As $x$ goes to infinity, $\frac{1}{x}$ approaches 0, so the function behaves like $x$ . Thus, as $x \to +\infty$ , $f(x) \to +\infty$ .
Limit as x approaches negative infinity: Similarly, as $x$ becomes infinitely negative, the numerator $x^2$ becomes positive and grows much faster than the denominator $x + 1$ , which becomes a large negative number. Using the rewritten function $f(x) = \frac{x}{1 + \frac{1}{x}}$ , when $x$ approaches negative infinity, the function tends toward $x$ , so the limit goes to negative infinity. Therefore, as $x \to -\infty$ , $f(x) \to -\infty$ .

So, to recap, as $x$ approaches -1, the limit does not exist, as $x$ approaches positive infinity, the limit is positive infinity, and as $x$ approaches negative infinity, the limit is negative infinity. Great job, guys! This is the most crucial part of this exercise.

Proving a Mathematical Statement

Now, let's move on to the fun part where we have to prove a mathematical statement. The statement we need to prove is: $(\forall x \in D_f)$ , $f(x) = x - 1 + \frac{1}{x+1}$ . Basically, we need to show that this equation holds true for all values of $x$ in the domain of $f$ . Remember that the domain excludes -1. This is a very cool concept, so let's get down to it. We need to do this step by step.

To prove this, we can start with the right-hand side of the equation, $x - 1 + \frac{1}{x+1}$ , and try to manipulate it to arrive at the left-hand side, $f(x) = \frac{x^2}{x+1}$ . Here's how we can do it:

Combine Terms: We can combine the terms by finding a common denominator, which is $(x + 1)$ .

$x - 1 + \frac{1}{x+1} = \frac{(x - 1)(x + 1)}{x + 1} + \frac{1}{x+1}$
Expand and Simplify: Expand the numerator $(x - 1)(x + 1)$ and then combine the fractions:

$\frac{(x - 1)(x + 1)}{x + 1} + \frac{1}{x+1} = \frac{x^2 - 1}{x + 1} + \frac{1}{x + 1}$

$\frac{x^2 - 1}{x + 1} + \frac{1}{x + 1} = \frac{x^2 - 1 + 1}{x + 1}$

$\frac{x^2 - 1 + 1}{x + 1} = \frac{x^2}{x + 1}$
Recognize the Result: The result is exactly $f(x) = \frac{x^2}{x+1}$ .

Therefore, we have shown that for all $x$ in the domain of $f$ , which excludes -1, the statement $f(x) = x - 1 + \frac{1}{x+1}$ is true. That was pretty neat, right? Great job!

Why is this important?

This kind of analysis is super important in mathematics and other fields. Understanding domains, limits, and being able to prove statements forms the basis of calculus, physics, and engineering. It allows us to model real-world phenomena, predict behaviors, and solve complex problems. You guys are doing great!

Additional Insights and Tips

Practice is Key: The more you practice these types of problems, the easier they become. Try working through similar examples to solidify your understanding. Doing more examples is the best thing you can do.
Visual Aids: Sketching the graph of the function can help you visualize the domain, limits, and other properties. Using a graphing calculator or online tool can be helpful for this.
Ask Questions: Don't hesitate to ask questions if you get stuck. Your teachers and fellow students are there to help you. Always feel free to ask questions. There's no shame in asking.
Break it Down: Break down complex problems into smaller, manageable steps. This will make the process less overwhelming.

Conclusion

Awesome work, everyone! We've successfully determined the domain of a function, calculated its limits, and proved a mathematical statement. You all did a great job. Keep up the hard work, and you'll become math wizards in no time. Keep practicing, and you will become experts! Math can be challenging, but with patience and effort, anyone can master it. Keep up the great work, and don't give up.

Hopefully, you now have a better grasp of function analysis, domains, limits, and mathematical proofs. Remember to always double-check your work, and always ask questions when you need help. Keep learning, and keep growing! This kind of exercise is very important, so be sure you study it! We covered a lot of important ground today. See you next time!