F(x) = (3x-5)/(x-2): In-Depth Function Analysis

Hey guys! Today, we're diving deep into the fascinating world of functions, specifically, the function f(x) = (3x - 5) / (x - 2). This might look a little intimidating at first, but trust me, by the end of this comprehensive analysis, you'll be a pro at understanding its ins and outs. We're going to break it down piece by piece, exploring everything from its domain and range to its asymptotes and behavior. So, grab your thinking caps, and let's get started!

1. Unveiling the Domain: Where Does Our Function Live?

First things first, let's talk about the domain. In simple terms, the domain of a function is the set of all possible input values (x-values) that will produce a valid output. Think of it as the function's natural habitat – the values where it can thrive and give us meaningful results. For our function, f(x) = (3x - 5) / (x - 2), we need to be mindful of one crucial rule: we can't divide by zero. Division by zero is a big no-no in the math world, as it leads to undefined results. So, we need to identify any x-values that would make the denominator (x - 2) equal to zero. Setting (x - 2) = 0, we quickly find that x = 2 is the culprit. This means that x = 2 is excluded from our domain. Therefore, the domain of f(x) is all real numbers except 2. We can express this mathematically in a few ways: using set notation, we can write it as {x | x ∈ ℝ, x ≠ 2}; using interval notation, we express it as (-∞, 2) ∪ (2, ∞). This interval notation tells us that our function happily accepts any x-value from negative infinity up to 2 (but not including 2), and then it picks up again immediately after 2 and continues to positive infinity. Visualizing this on a number line, we would see a line stretching endlessly in both directions, with a hole punched out at x = 2. Understanding the domain is a fundamental step because it sets the stage for everything else we'll explore about the function. It tells us where our function is defined and where we need to be cautious. We'll see how this knowledge becomes crucial when we analyze asymptotes and the overall behavior of the function. So, with the domain securely in our grasp, let's move on to the next exciting aspect: the range!

2. Exploring the Range: What Values Can Our Function Achieve?

Now that we've nailed down the domain, let's shift our focus to the range. The range, in contrast to the domain, is the set of all possible output values (y-values or f(x) values) that our function can produce. It's like asking,