Cheetah's Speed Solving And Interpreting Function Operations

How to solve and interpret the composite function (f/g)(3)?

In the captivating realm of mathematics, we often encounter functions that serve as powerful tools for modeling real-world phenomena. These functions, with their ability to represent relationships between variables, allow us to gain deeper insights into the intricacies of the world around us. Today, we embark on a mathematical journey to explore the fascinating world of cheetahs, the fastest land animals on Earth, and unravel the secrets of their incredible speed using the language of functions.

In this exploration, we will delve into the realm of function operations, specifically focusing on the quotient of two functions. By understanding how to combine functions through division, we can extract valuable information about the cheetah's rate of travel, which is a crucial aspect of its hunting prowess and survival in the African savanna. So, let's prepare to unleash our mathematical prowess and uncover the cheetah's secrets, one function at a time.

Problem Unraveling the Mathematical Puzzle

Let's begin by laying out the mathematical framework for our cheetah-themed exploration. We are presented with two functions, each representing a distinct aspect of the cheetah's journey:

• The function $f(x) = 6x + 8$ elegantly captures the distance, measured in miles, that our cheetah friend covers during its run. Here, the variable $x$ represents the time, measured in hours, that the cheetah has been running.
• On the other hand, the function $g(x) = x - 2$ focuses on the time aspect of the cheetah's journey, specifically the time the cheetah ran, also measured in hours.

Our mission, should we choose to accept it, is to embark on a mathematical quest to solve $\left(\frac{f}{g}\right)(3)$. This expression, at first glance, might seem like a cryptic code, but fear not, for we shall decipher its meaning and unlock the secrets it holds.

But our journey doesn't end with merely finding the numerical solution. We must also don our interpretive hats and unravel the real-world significance of our answer. What does this numerical value tell us about the cheetah's incredible speed? What insights can we glean about its hunting strategies and survival instincts?

So, let's prepare to embark on this mathematical adventure, where we will combine the power of functions with the captivating world of cheetahs. Together, we shall unravel the mysteries of their speed and gain a deeper appreciation for the beauty of mathematics in action.

Solution Deciphering the Mathematical Code

Now, let's delve into the heart of the problem and embark on the journey of finding the solution. Our goal is to solve $\left(\frac{f}{g}\right)(3)$, which involves the quotient of two functions, $f(x)$ and $g(x)$, evaluated at the specific value of $x = 3$.

To conquer this mathematical challenge, we will break it down into a series of manageable steps, each building upon the previous one. First, we need to understand what $\left(\frac{f}{g}\right)(x)$ actually means. It represents a new function formed by dividing the function $f(x)$ by the function $g(x)$. In mathematical notation, we can express this as:

$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$

Now that we have a clear understanding of the quotient of functions, we can substitute the given functions, $f(x) = 6x + 8$ and $g(x) = x - 2$, into the expression:

$\left(\frac{f}{g}\right)(x) = \frac{6x + 8}{x - 2}$

With the quotient function in hand, we are ready to take the final step: evaluating it at $x = 3$. This means we will substitute $x = 3$ into the expression we just derived:

$\left(\frac{f}{g}\right)(3) = \frac{6(3) + 8}{3 - 2}$

Now, it's simply a matter of performing the arithmetic operations. Let's start with the numerator:

$6(3) + 8 = 18 + 8 = 26$

And the denominator:

$3 - 2 = 1$

Finally, we divide the numerator by the denominator:

$\frac{26}{1} = 26$

Thus, we have arrived at the solution: $\left(\frac{f}{g}\right)(3) = 26$. But our journey doesn't end here. We must now interpret this numerical result in the context of the cheetah's journey.

Interpretation Unraveling the Cheetah's Tale

Now comes the moment of truth: interpreting the numerical solution we obtained in the context of the cheetah's journey. We found that $\left(\frac{f}{g}\right)(3) = 26$. But what does this number signify in the real world?

Recall that $f(x)$ represents the distance the cheetah runs in miles, and $g(x)$ represents the time the cheetah runs in hours. Therefore, the quotient $\frac{f(x)}{g(x)}$ represents the cheetah's rate of travel, which is the distance traveled per unit of time.

In our specific case, we evaluated this rate at $x = 3$ hours. So, the value 26 represents the cheetah's rate of travel at 3 hours. Since the distance is measured in miles and the time in hours, the rate is expressed in miles per hour.

Therefore, we can confidently state that 26 is the cheetah's rate in miles per hour after running for 3 hours. This provides us with a crucial insight into the cheetah's incredible speed and its ability to cover vast distances in a relatively short amount of time.

Imagine the cheetah sprinting across the African savanna, its powerful legs propelling it forward at an astonishing speed. After 3 hours of running, it is covering ground at a rate of 26 miles per hour. This remarkable speed allows the cheetah to effectively hunt prey, evade predators, and navigate its challenging environment.

The function operations we performed have unveiled a key aspect of the cheetah's behavior, its speed. By understanding the relationship between distance, time, and rate, we gain a deeper appreciation for the mathematical principles that govern the natural world. Our journey into the world of functions has not only provided us with a numerical solution but also a richer understanding of the cheetah's remarkable abilities.

Conclusion The Power of Functions in Action

In this mathematical exploration, we embarked on a journey to unravel the secrets of the cheetah's speed using the language of functions. We were presented with two functions, one representing the distance the cheetah runs and the other representing the time it runs. Our mission was to solve $\left(\frac{f}{g}\right)(3)$ and interpret the answer in the context of the cheetah's journey.

Through a step-by-step approach, we successfully navigated the realm of function operations. We learned that $\left(\frac{f}{g}\right)(x)$ represents the quotient of two functions, formed by dividing $f(x)$ by $g(x)$. We then substituted the given functions and evaluated the expression at $x = 3$, arriving at the numerical solution of 26.

But our journey didn't end with just a number. We delved into the interpretation of this result, recognizing that it represents the cheetah's rate of travel in miles per hour. After 3 hours of running, the cheetah is covering ground at an impressive rate of 26 miles per hour. This insight allowed us to appreciate the cheetah's remarkable speed and its crucial role in its survival.

This exploration serves as a powerful reminder of the ability of mathematics to model and interpret real-world phenomena. Functions, in particular, provide a versatile tool for representing relationships between variables and gaining deeper insights into the world around us. By combining mathematical concepts with real-world scenarios, we can unlock a richer understanding of the natural world and the intricate processes that govern it.

So, the next time you marvel at the speed of a cheetah, remember the mathematical journey we undertook. We have seen firsthand how the language of functions can illuminate the cheetah's incredible speed and provide a window into the fascinating world of mathematical modeling.

• Functions: Mathematical relationships that map inputs to outputs, essential for modeling real-world phenomena.
• Function Operations: Combining functions through addition, subtraction, multiplication, division, and composition, enabling complex modeling.
• Quotient of Functions: Dividing one function by another, often representing rates or ratios.
• Rate of Travel: Distance traveled per unit of time, crucial for understanding motion and speed.
• Mathematical Modeling: Using mathematical concepts to represent and analyze real-world situations.
• How to solve the composite function (f/g)(3)?
• How to interpret the composite function (f/g)(3)?