Unveiling F(x+8) And G(x/4) A Mathematical Exploration
Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of functions. We've got two intriguing characters to introduce: f(x) = x² - 4x - 6 and g(x) = (3x - 5) / (x - 4). Our mission, should we choose to accept it, is to unravel the mysteries of f(x+8) and g(x/4). Buckle up, because this is going to be a fun ride!
Cracking the Code of f(x+8)
Let's kick things off with f(x+8). The key here is function composition, which might sound fancy, but it's actually quite straightforward. We're essentially replacing every 'x' in the original function f(x) with '(x+8)'. Think of it like substituting ingredients in a recipe – we're swapping 'x' for '(x+8)'. So, where do we even start with functions? Well, let's lay the groundwork by restating our original function: f(x) = x² - 4x - 6. The core idea behind this function is that whatever input we give it (in this case, 'x'), the function will square it, subtract four times the input, and then subtract six. Now, instead of feeding 'x' into this machine, we're going to feed in '(x+8)'. This is where the magic happens! When substituting (x+8) into f(x), remember the order of operations – parentheses, exponents, multiplication, and then addition/subtraction. Let's start by replacing every instance of 'x' in our original function with '(x+8)': f(x+8) = (x+8)² - 4(x+8) - 6. Now, we need to simplify this expression. The first step is to expand (x+8)². Remember that squaring a binomial means multiplying it by itself: (x+8)² = (x+8)(x+8). Using the FOIL method (First, Outer, Inner, Last), we get: (x+8)(x+8) = x² + 8x + 8x + 64 = x² + 16x + 64. So, f(x+8) now looks like this: f(x+8) = x² + 16x + 64 - 4(x+8) - 6. Next, we need to distribute the -4 across the (x+8) term: -4(x+8) = -4x - 32. Now our expression is: f(x+8) = x² + 16x + 64 - 4x - 32 - 6. Finally, we combine like terms to simplify: x² terms: We only have one x² term, so it remains as x². x terms: We have 16x and -4x, which combine to 12x. Constant terms: We have 64, -32, and -6, which combine to 26. Putting it all together, we get the simplified expression for f(x+8): f(x+8) = x² + 12x + 26. And there you have it! We've successfully cracked the code of f(x+8), transforming it from a seemingly complex expression into a much more manageable quadratic. This process highlights the power of algebraic manipulation and the importance of carefully applying the order of operations. Next up, we'll tackle g(x/4), which will present its own unique set of challenges and insights. Stay tuned as we continue our mathematical adventure!
The Substitution Step
The first step is to substitute (x+8) into the function. This gives us:
f(x+8) = (x+8)² - 4(x+8) - 6
Expanding and Simplifying
Now, let's expand and simplify this expression. Remember to use the order of operations (PEMDAS/BODMAS):
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Expand (x+8)²:
(x+8)² = (x+8)(x+8) = x² + 16x + 64
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Distribute the -4:
-4(x+8) = -4x - 32
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Rewrite the expression:
f(x+8) = x² + 16x + 64 - 4x - 32 - 6
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Combine like terms:
f(x+8) = x² + (16x - 4x) + (64 - 32 - 6)
f(x+8) = x² + 12x + 26
So, after all the algebraic maneuvering, we've arrived at the simplified form of f(x+8): f(x+8) = x² + 12x + 26. This quadratic equation is a lot cleaner than our initial expression, isn't it? This transformation showcases the power of algebraic manipulation to turn complex expressions into simpler, more manageable forms. It's like taking a tangled knot and patiently unraveling it to reveal a smooth, continuous strand. But our journey doesn't end here! We've still got another function to conquer: g(x/4). This one might throw us a curveball or two, but we're well-equipped to handle it. Remember, the key to success in these kinds of problems is to break them down into smaller, more manageable steps. We've already seen how substitution, expansion, and simplification can work wonders, and we'll be using those same tools as we tackle g(x/4). So, let's keep our mathematical gears turning and get ready for the next challenge!
Decoding g(x/4)
Alright, let's shift our focus to g(x/4). This time, we're dealing with a rational function, which means we have a fraction in the mix. Our original function is g(x) = (3x - 5) / (x - 4). Just like before, we're going to replace every 'x' in this function with 'x/4'. This might seem a bit trickier because of the fraction, but don't worry, we'll take it step by step.
The Substitution Tango
First, we substitute 'x/4' into g(x):
g(x/4) = (3(x/4) - 5) / ((x/4) - 4)
Now, this looks a bit messy, doesn't it? We've got fractions within fractions, which is never a pretty sight. But we're not going to let that intimidate us. We're going to clean this up using our trusty algebraic tools.
Simplifying the Complex Fraction
Our goal is to get rid of the fractions in the numerator and the denominator. To do this, we can multiply both the numerator and the denominator by the least common multiple (LCM) of the denominators of the inner fractions. In this case, the only denominator we need to worry about is 4. So, we'll multiply both the top and bottom of our fraction by 4. When tackling a complex fraction like this, it's crucial to remember the fundamental principle: multiplying the numerator and the denominator by the same non-zero value doesn't change the overall value of the fraction. This is because we're essentially multiplying by 1, albeit in a disguised form. This technique allows us to manipulate the expression without altering its essence, making it a powerful tool in our algebraic arsenal. In the context of simplifying g(x/4), multiplying by 4 will help us eliminate the fractions within the larger fraction, making the expression much easier to handle and ultimately leading us to the simplified form we're after. The ability to recognize and apply such simplification techniques is a hallmark of mathematical fluency, allowing us to navigate complex expressions with confidence and precision. Remember, practice makes perfect, so the more you work with these kinds of problems, the more comfortable you'll become with the various strategies for simplification.
g(x/4) = 4 * (3(x/4) - 5) / 4 * ((x/4) - 4)
Distributing and Tidying Up
Now, we distribute the 4 in both the numerator and the denominator:
g(x/4) = (4 * 3(x/4) - 4 * 5) / (4 * (x/4) - 4 * 4)
g(x/4) = (3x - 20) / (x - 16)
And there we have it! We've successfully decoded g(x/4) and simplified it to g(x/4) = (3x - 20) / (x - 16). This rational function is now in a much cleaner form, free from those pesky fractions within fractions. This journey through simplifying g(x/4) has highlighted the importance of recognizing and addressing complex fractions. By strategically multiplying the numerator and denominator by the LCM, we were able to eliminate the inner fractions and arrive at a more elegant expression. This technique is a valuable addition to your mathematical toolkit, ready to be deployed whenever you encounter similar challenges. As we wrap up this exploration of f(x+8) and g(x/4), it's worth reflecting on the core concepts we've employed. Function composition, substitution, expansion, simplification, and dealing with fractions – these are all fundamental skills in algebra and calculus. Mastering these concepts will not only help you solve problems like this one, but also lay a strong foundation for more advanced mathematical pursuits.
Final Thoughts
So, after our mathematical escapade, we've discovered that:
f(x+8) = x² + 12x + 26
g(x/4) = (3x - 20) / (x - 16)
Great job, everyone! We successfully navigated the world of function composition and simplification. Remember, math is an adventure, and every problem is a new puzzle to solve.