Function F(x)=8/(x-1): Domain & Asymptotes

Hey math enthusiasts! Today, we're diving deep into the nitty-gritty of a seemingly simple rational function: $f(x) = \frac{8}{x-1}$. Don't let its straightforward appearance fool you, guys. Understanding the domain, vertical asymptotes, and horizontal asymptotes of functions like this is absolutely crucial for grasping their behavior, graphing them accurately, and solving more complex problems down the line. So, buckle up, because we're about to break it all down in a way that's easy to digest and, dare I say, even fun!

Demystifying the Domain: Where Can Our Function Hang Out?

Alright, let's start with the domain. Think of the domain as the set of all possible input values (the 'x' values) for which our function is defined and makes mathematical sense. For rational functions, which are basically fractions with polynomials in the numerator and denominator, the biggest troublemaker is the denominator. Why? Because you can never, ever divide by zero! It's like trying to share a pizza with zero people – it just doesn't compute, mathematically speaking. So, for our function $f(x) = \frac{8}{x-1}$, we need to figure out which 'x' values would make that denominator, $x-1$, equal to zero. Setting $x-1 = 0$ is super easy to solve – just add 1 to both sides, and boom, you get $x = 1$. This means that when $x$ is exactly 1, our function is undefined. It's a no-go zone! Every other real number, though? Totally fine. We can plug them in, and we'll get a valid output. So, the domain of $f(x)$ is all real numbers except for 1. We can express this in a few cool ways. In set notation, it looks like this: $\{x \in \mathbb{R} \mid x \neq 1\}$. If you prefer interval notation, which is super handy for graphing, it's $(-\infty, 1) \cup (1, \infty)$. See that little 'U'? That means 'union', just combining two separate intervals. It's like saying, 'We can have any number less than 1, OR any number greater than 1.' So, to recap, the domain tells us precisely where our function lives and where it doesn't. It's the first big piece of the puzzle in understanding our function's personality.

Hunting for Vertical Asymptotes: The Forbidden Zones

Now, let's talk about vertical asymptotes. These are like invisible walls that our function's graph can get infinitely close to, but never actually touch or cross. They occur at the x-values where the function becomes undefined, specifically due to the denominator becoming zero, provided that the numerator doesn't also become zero at the same spot (if both are zero, it's usually a hole, not an asymptote, but we'll save that for another day!). Remember how we found that $x=1$ makes our denominator zero? That's our prime suspect for a vertical asymptote! Since the numerator (which is just the constant 8) is definitely not zero when $x=1$, we have ourselves a bona fide vertical asymptote. The line $x = 1$ is this vertical boundary. What does this mean graphically? As 'x' gets closer and closer to 1 (from the left side, like 0.9, 0.99, 0.999, or from the right side, like 1.1, 1.01, 1.001), the 'y' value (the output of our function) will shoot off towards positive infinity or negative infinity. It's like the function is saying, 'Whoa, almost there, but nope, can't touch this line!' Finding vertical asymptotes is all about spotting those x-values that zero out the denominator without zeroing out the numerator. They are super important because they define the boundaries of where the function's behavior gets wild and unbounded. For $f(x) = \frac{8}{x-1}$, the vertical asymptote is straightforwardly at $x=1$. It's a key feature that helps us sketch the graph and understand how the function behaves in its vicinity. Pretty neat, right? We're uncovering the structure of this function piece by piece!

Spotting Horizontal Asymptotes: The Long-Term Trend

Finally, let's zoom out and consider the horizontal asymptotes. While vertical asymptotes describe what happens when 'x' gets close to a specific number, horizontal asymptotes describe the function's behavior as 'x' heads off towards positive or negative infinity. In simpler terms, what value does our function's output 'y' approach as 'x' gets ridiculously large (positive) or ridiculously small (negative)? For rational functions, like our $f(x) = \frac{8}{x-1}$, there are a few rules of thumb based on the degrees of the polynomials in the numerator and the denominator. Let's say our function is in the general form $f(x) = \frac{a ">⁢</ x^m + ext{lower degree terms}}{b ">⁢</ x^n + ext{lower degree terms}}$.

1. If the degree of the numerator (m) is LESS THAN the degree of the denominator (n): The horizontal asymptote is always $y = 0$. This is our situation! In $f(x) = \frac{8}{x-1}$, the numerator is 8 (which we can think of as $8x^0$, so the degree is 0), and the denominator is $x-1$ (the degree is 1). Since $0 < 1$, the rule applies, and our horizontal asymptote is $y = 0$. This means as 'x' gets super big (positive or negative), the function's value gets closer and closer to zero.

2. If the degree of the numerator (m) is EQUAL TO the degree of the denominator (n): The horizontal asymptote is the ratio of the leading coefficients, $y = \frac{a}{b}$.

3. If the degree of the numerator (m) is GREATER THAN the degree of the denominator (n): There is no horizontal asymptote. The function will grow (or decrease) without bound.

So, for $f(x) = \frac{8}{x-1}$, since the degree of the numerator (0) is less than the degree of the denominator (1), the horizontal asymptote is indeed $y = 0$. This tells us about the long-term trend of our function. As 'x' goes to $\infty$ or $-\infty$, the graph of $f(x)$ will get closer and closer to the x-axis, but it might never actually touch it (though it can cross a horizontal asymptote in the middle of the graph, unlike a vertical one). It's all about the end behavior, guys!

Putting It All Together: The Full Picture

So, let's summarize what we've uncovered about $f(x) = \frac{8}{x-1}$:

• Domain: All real numbers except $x=1$. In interval notation: $(-\infty, 1) \cup (1, \infty)$. This means our function is defined everywhere except at $x=1$.
• Vertical Asymptote(s): $x = 1$. This is the vertical line that the graph approaches but never touches as 'x' nears 1.
• Horizontal Asymptote: $y = 0$. This is the horizontal line (the x-axis) that the graph approaches as 'x' heads towards positive or negative infinity.

Understanding these three key components—the domain, vertical asymptotes, and horizontal asymptotes—gives you a fantastic blueprint for sketching the graph of a rational function and understanding its fundamental behavior. It's like getting the X-ray vision for these types of functions! Keep practicing, and you'll be spotting these features like a pro in no time. Happy graphing!