Simplify & Discuss: 1/(1-x) + 2/(1+x)

Hey guys! Today, we're diving deep into a fascinating mathematical expression: 1/(1-x) + 2/(1+x). This might look intimidating at first glance, but trust me, we're going to break it down step by step, making it super easy to understand. We'll explore how to simplify it, discuss its properties, and even look at some real-world applications. So, buckle up and get ready to unleash your inner mathematician!

Simplifying the Expression: Making Math Magic

The core of understanding any mathematical expression lies in simplifying it. Our expression, 1/(1-x) + 2/(1+x), can be simplified by finding a common denominator. This is a fundamental technique in algebra, and it's crucial for manipulating fractions effectively. Think of it like combining slices of different pizzas – you need to cut them into the same size pieces before you can accurately count the total. So, let’s get started with the simplification process. First, we identify the denominators: (1-x) and (1+x). The least common denominator (LCD) is simply the product of these two, which is (1-x)(1+x). Now, we rewrite each fraction with this common denominator. The first fraction, 1/(1-x), needs to be multiplied by (1+x)/(1+x). This gives us (1*(1+x))/((1-x)(1+x)), which simplifies to (1+x)/((1-x)(1+x)). Similarly, the second fraction, 2/(1+x), needs to be multiplied by (1-x)/(1-x). This yields (2*(1-x))/((1+x)(1-x)), which simplifies to (2-2x)/((1-x)(1+x)). Now that both fractions have the same denominator, we can add them together. We combine the numerators: (1+x) + (2-2x). This simplifies to 3-x. So, the numerator is now 3-x, and the denominator remains (1-x)(1+x). The expression now looks like (3-x)/((1-x)(1+x)). But wait, we can simplify the denominator further! Notice that (1-x)(1+x) is a difference of squares. Remember that (a-b)(a+b) = a² - b²? Applying this identity, we get (1-x)(1+x) = 1² - x² = 1 - x². Therefore, our fully simplified expression is (3-x)/(1-x²). Isn't that neat? We've taken a seemingly complex expression and boiled it down to something much more manageable. This simplified form is not only easier to work with, but it also reveals important properties of the original expression, which we'll explore in the next sections.

Exploring the Properties: Unveiling the Expression's Personality

Now that we've simplified our expression, (3-x)/(1-x²), it's time to put on our detective hats and explore its properties. Think of these properties as the expression's personality traits – they tell us how it behaves under different conditions. One of the most important properties to investigate is the domain of the expression. The domain is the set of all possible input values (x-values) for which the expression is defined. In other words, it's where our expression is