Determining The Nature Of F(8 + √3) For F(x) = 3x² - 7
Hey guys! Today, we're diving deep into the fascinating world of functions, specifically exploring the nature of f(8 + √3) when our function f(x) is defined as 3x² - 7. This might seem a bit daunting at first, but trust me, we'll break it down step-by-step and make it super easy to understand. So, buckle up and let's embark on this mathematical adventure together!
Understanding the Function and the Challenge
In this section, we will start by understanding the given function. Our mission is to determine the nature of f(8 + √3), and the function we're working with is f(x) = 3x² - 7. What does this mean? Well, a function is essentially a mathematical machine. You feed it an input (in this case, 'x'), and it spits out an output based on a specific rule. Our rule here is: take the input (x), square it (x²), multiply the result by 3 (3x²), and then subtract 7 (3x² - 7). So, if we were to plug in a simple number like 2 into this function, we'd get f(2) = 3(2)² - 7 = 3(4) - 7 = 12 - 7 = 5. Easy peasy, right?
Now, the challenge lies in the input we're dealing with: 8 + √3. It's not just a simple whole number; it's a combination of a whole number (8) and an irrational number (√3). Irrational numbers are those that cannot be expressed as a simple fraction (like 1/2 or 3/4); their decimal representations go on forever without repeating. √3 is a classic example, approximately equal to 1.73205.... So, plugging 8 + √3 directly into our function might seem a bit intimidating. But don't worry, we're going to tackle it systematically.
The core question we're trying to answer is: what kind of number will f(8 + √3) be? Will it be a whole number? A fraction? An irrational number? Or something else entirely? To answer this, we need to carefully substitute 8 + √3 for 'x' in our function and then simplify the expression. This is where our algebraic skills will come in handy. We'll need to remember how to square binomials (expressions with two terms) and how to deal with square roots. But with a bit of patience and attention to detail, we'll unravel the mystery of f(8 + √3) and reveal its true nature.
Step-by-Step Evaluation of f(8 + √3)
Alright, let's get our hands dirty and dive into the step-by-step evaluation of f(8 + √3). Remember our function, f(x) = 3x² - 7? Our mission is to replace 'x' with (8 + √3) and simplify the resulting expression. So, let's do it!
First, we substitute (8 + √3) for 'x' in the function: f(8 + √3) = 3(8 + √3)² - 7. Now, we need to tackle the squared term: (8 + √3)². This means we're multiplying (8 + √3) by itself: (8 + √3)(8 + √3). To do this, we'll use the FOIL method (First, Outer, Inner, Last) or the distributive property. Let's break it down:
- First: 8 * 8 = 64
- Outer: 8 * √3 = 8√3
- Inner: √3 * 8 = 8√3
- Last: √3 * √3 = 3
Now, we add these terms together: 64 + 8√3 + 8√3 + 3. We can combine the like terms (the whole numbers and the terms with √3): 64 + 3 = 67 and 8√3 + 8√3 = 16√3. So, (8 + √3)² simplifies to 67 + 16√3. Great! We've conquered the squared term.
Now, let's plug this back into our original expression: f(8 + √3) = 3(67 + 16√3) - 7. Next, we need to distribute the 3 across the terms inside the parentheses: 3 * 67 = 201 and 3 * 16√3 = 48√3. So, we have f(8 + √3) = 201 + 48√3 - 7. Finally, we combine the like terms (the whole numbers): 201 - 7 = 194. This gives us our final result: f(8 + √3) = 194 + 48√3.
We've successfully evaluated f(8 + √3)! But what does this result tell us about its nature? Let's explore that in the next section.
Determining the Nature of the Result: Rational or Irrational?
Okay, guys, we've arrived at the exciting part where we determine the nature of our result: f(8 + √3) = 194 + 48√3. Is it a rational number, an irrational number, or something else entirely? To answer this, we need to understand the properties of rational and irrational numbers.
Rational numbers, as we briefly mentioned earlier, are numbers that can be expressed as a fraction p/q, where p and q are integers (whole numbers) and q is not zero. Examples include 2 (which can be written as 2/1), -3/4, and 0.5 (which can be written as 1/2). Rational numbers have decimal representations that either terminate (like 0.5) or repeat in a pattern (like 0.333...).
Irrational numbers, on the other hand, cannot be expressed as a simple fraction. Their decimal representations go on forever without repeating. We already encountered √3, a classic example of an irrational number. Other famous irrational numbers include π (pi) and the square root of any non-perfect square (like √2, √5, √7, etc.).
Now, let's analyze our result: 194 + 48√3. We have two terms: 194 and 48√3. 194 is a whole number, and therefore it's rational. But what about 48√3? Here's where it gets interesting. We know that √3 is irrational. When we multiply a rational number (48) by an irrational number (√3), the result (48√3) is also irrational.
Finally, we're adding a rational number (194) to an irrational number (48√3). The sum of a rational number and an irrational number is always irrational. Think about it this way: the irrational part (48√3) has a decimal representation that goes on forever without repeating. Adding a rational number (194) to it won't suddenly make the decimal representation terminate or repeat; it will still go on forever without repeating. Therefore, the entire expression, 194 + 48√3, is irrational.
So, the final verdict is in: f(8 + √3) is an irrational number. We've successfully navigated the substitution, simplification, and analysis to arrive at this conclusion. Pat yourselves on the back, guys! You've tackled a challenging problem and emerged victorious.
Conclusion: The Intriguing Nature of Function Outputs
Wohoo! We've reached the conclusion of our mathematical journey, and what a journey it has been! We set out to determine the nature of f(8 + √3) for the function f(x) = 3x² - 7, and we've successfully unraveled the mystery. By carefully substituting, simplifying, and analyzing, we discovered that f(8 + √3) = 194 + 48√3, which is an irrational number.
This exploration highlights the fascinating interplay between functions, rational numbers, and irrational numbers. It demonstrates that even when we plug in a seemingly complex input (like 8 + √3) into a relatively simple function (like 3x² - 7), the output can have an intriguing nature. It also reinforces the importance of understanding the properties of different types of numbers, like rational and irrational numbers, in order to fully grasp the behavior of functions.
More broadly, this exercise exemplifies the power of mathematical thinking. By breaking down a complex problem into smaller, manageable steps, we were able to solve it systematically. We used our algebraic skills to simplify expressions, our knowledge of number properties to analyze the result, and our logical reasoning to draw a conclusion. These are valuable skills that can be applied not only in mathematics but also in many other areas of life.
So, what's the takeaway, guys? Mathematics isn't just about memorizing formulas and procedures; it's about developing a way of thinking that allows us to approach challenges with confidence and clarity. By embracing the process of exploration, questioning, and problem-solving, we can unlock a deeper understanding of the world around us. And who knows, maybe we'll even uncover a few more mathematical mysteries along the way!
