Find The Horizontal Asymptote Of F(x)=8x/(48x+7)

Dec 9, 2025 by Admin 49 views

Hey math whizzes! Today, we're diving deep into the fascinating world of functions and their graphical behavior. Specifically, we're going to tackle the concept of horizontal asymptotes and figure out exactly what they are for the function $f(x)= rac{8 x}{48 x+7}$ . Now, I know "asymptote" might sound like a big, scary word, but trust me, once you get the hang of it, it's pretty straightforward. Think of a horizontal asymptote as a line that the graph of your function gets really, really close to as the input values (the x-values) go off to infinity or negative infinity. It's like a guiding line, showing you where the function is headed in the long run. We're going to break down exactly how to find this elusive line for our specific function, $f(x)= rac{8 x}{48 x+7}$ . So, grab your calculators, maybe a cup of coffee, and let's get started on this mathematical adventure!

Understanding Horizontal Asymptotes

Alright guys, let's get down to the nitty-gritty of what a horizontal asymptote actually is. In simple terms, it's a horizontal line, usually represented by $y=c$ (where 'c' is some constant number), that describes the end behavior of a function. This means we're looking at what happens to the function's output (the y-value) as the input (the x-value) gets super large, either in the positive direction (approaching positive infinity, $\infty$ ) or the negative direction (approaching negative infinity, $-\infty$ ). So, if the limit of $f(x)$ as $x$ approaches $\infty$ is a specific number 'c', then the line $y=c$ is a horizontal asymptote. Similarly, if the limit of $f(x)$ as $x$ approaches $-\infty$ is also 'c', we still only have one horizontal asymptote at $y=c$ . It's crucial to remember that the graph of a function can cross its horizontal asymptote, but it will generally approach it as x gets larger and larger in magnitude. Think of it like a race car on a track; it might swerve a bit, but it's generally sticking to the general path of the track. The function $f(x)= rac{8 x}{48 x+7}$ is a rational function, which means it's a ratio of two polynomials. Rational functions are prime candidates for having horizontal asymptotes, and understanding their structure is key to finding them. We'll be using the concept of limits to determine the behavior of our function as x heads towards infinity. This involves looking at the highest powers of x in the numerator and the denominator, which is a super handy shortcut once you know the rules. So, buckle up, because we're about to unlock the secret to identifying these important features of our function's graph!

Finding the Horizontal Asymptote for f(x) = (8x)/(48x+7)

Now, let's get our hands dirty and actually calculate the horizontal asymptote for our specific function, $f(x)= rac{8 x}{48 x+7}$ . The most common and effective way to find a horizontal asymptote for a rational function like this is to examine the degrees of the polynomial in the numerator and the denominator. Remember, the degree of a polynomial is simply the highest power of the variable (in this case, x) present in that polynomial. For our function, $f(x)= rac{8 x}{48 x+7}$ , the numerator is $8x$ (which has a degree of 1, since $x = x^1$ ) and the denominator is $48x+7$ (which also has a degree of 1). When the degrees of the numerator and the denominator are the same, as they are in this case (both are 1), the horizontal asymptote is the line $y = rac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$ . The leading coefficient is simply the number multiplying the highest power of x. In our numerator, $8x$ , the leading coefficient is 8. In our denominator, $48x+7$ , the leading coefficient is 48. Therefore, the horizontal asymptote for $f(x)= rac{8 x}{48 x+7}$ is $y = rac{8}{48}$ .

Simplifying the Result

We've done the heavy lifting, guys, and found that the horizontal asymptote is $y = rac{8}{48}$ . But we're not quite done yet! Just like simplifying a fraction makes it easier to understand, simplifying the equation of our asymptote makes it cleaner and more digestible. So, let's take that fraction $\frac{8}{48}$ and simplify it. Both 8 and 48 are divisible by their greatest common divisor, which is 8. If we divide both the numerator and the denominator by 8, we get:

$\frac{8 \div 8}{48 \div 8} = \frac{1}{6}$

So, the simplified equation for our horizontal asymptote is $y = \frac{1}{6}$ . This means that as the x-values in our function $f(x)= rac{8 x}{48 x+7}$ get incredibly large (approaching positive or negative infinity), the y-values of the function will get closer and closer to $\frac{1}{6}$ . It's the line that the graph of our function essentially hugs as it travels off into the distant reaches of the coordinate plane. Pretty neat, huh? Always remember to simplify your results whenever possible; it's good practice and makes your answers look super professional!

Alternative Method: Dividing by the Highest Power of x

Let's explore another super cool method to find the horizontal asymptote of $f(x)= rac{8 x}{48 x+7}$ . This technique involves dividing every term in the numerator and the denominator by the highest power of x present in the entire fraction. In our case, the highest power of x is $x^1$ (or just x). So, we'll divide each term by x:

f(x) = \frac{\frac{8x}{x}}{\frac{48x}{x} + \frac{7}{x}}

Now, let's simplify each part:

$\frac{8x}{x} = 8$
$\frac{48x}{x} = 48$
$\frac{7}{x}$ remains as $\frac{7}{x}$

So, our function transforms into:

f(x) = \frac{8}{48 + \frac{7}{x}}

Now, here's the magic trick: we need to consider what happens as x approaches infinity ( $\infty$ ) or negative infinity ( $-\infty$ ). As x gets incredibly large, the term $\frac{7}{x}$ gets incredibly small, approaching 0. Think about it: 7 divided by a gigantic number is practically zero! So, as $x \to \infty$ or $x \to -\infty$ , the expression $\frac{7}{x} \to 0$ .

Substituting this back into our transformed function, we get:

f(x) \approx \frac{8}{48 + 0} = \frac{8}{48}

And again, we simplify this fraction to get $\frac{1}{6}$ . Therefore, the horizontal asymptote is $y = \frac{1}{6}$ . This method reinforces our previous finding and showcases a more formal way to think about limits and asymptotes. It's a great way to build intuition about how functions behave at extremes. Keep practicing these methods, guys, and you'll be an asymptote expert in no time!

What if the Degrees Were Different?

It's super important to know what happens when the degrees of the numerator and denominator aren't the same, because that changes how we find the horizontal asymptote. We've seen the case where the degrees are equal, but let's quickly touch upon the other scenarios. Scenario 1: Degree of the numerator is LESS than the degree of the denominator. For example, if we had a function like $g(x) = \frac{x+2}{x^2+5}$ . Here, the degree of the numerator is 1, and the degree of the denominator is 2. When the numerator's degree is smaller, the horizontal asymptote is always the line $y=0$ . This is because as x gets huge, the denominator grows much faster than the numerator, making the entire fraction shrink towards zero. Scenario 2: Degree of the numerator is GREATER than the degree of the denominator. Consider a function like $h(x) = \frac{x^2+1}{x+3}$ . Here, the degree of the numerator (2) is greater than the degree of the denominator (1). In this situation, there is no horizontal asymptote. Instead, the function might have a slant (or oblique) asymptote, which is a diagonal line that the function approaches. Finding slant asymptotes involves polynomial long division, which is a whole other fun topic! So, for our original function $f(x)= rac{8 x}{48 x+7}$ , we were lucky because the degrees were equal, leading us directly to a nice, clean horizontal asymptote. Understanding these different degree comparisons is key to mastering asymptotes for all sorts of rational functions. It's like having a cheat sheet for different types of function behavior!

Conclusion: The Horizontal Asymptote of f(x)

So, after all that mathematical exploration, we've definitively found the horizontal asymptote for the function $f(x)= rac{8 x}{48 x+7}$ . By comparing the degrees of the numerator and the denominator (which were both 1), and then taking the ratio of their leading coefficients, we arrived at the equation $y = rac{8}{48}$ . We then took the extra step to simplify this fraction, giving us the final, elegant answer: $y = \frac{1}{6}$ . This means that as the input values for our function get extremely large in either the positive or negative direction, the output values will get progressively closer and closer to $\frac{1}{6}$ . It's the invisible line guiding the function's behavior at the extremes. Remember this process, guys: compare degrees, and if they're equal, take the ratio of leading coefficients. Don't forget to simplify! Mastering horizontal asymptotes is a fundamental skill in understanding the graphs of rational functions, and with a little practice, you'll be spotting them like a pro. Keep exploring, keep questioning, and keep enjoying the awesome world of mathematics!