Finding The Value Of A Given Composite Functions F(x) And G(x)

Hey guys! Ever stumbled upon a math problem that looks like a puzzle? Well, today we're diving into one involving composite functions. Specifically, we're going to figure out how to find the value of a variable when we're given two functions, f(x) and g(x), and the result of their composition. This might sound intimidating, but trust me, we'll break it down step by step. Let's get started!

Understanding Composite Functions

Before we jump into solving the problem, let's make sure we're all on the same page about what a composite function actually is. Imagine you have two machines: one that doubles a number and adds three, and another that squares a number, subtracts three times the number, and then adds five. These machines represent our functions, f(x) and g(x), respectively. A composite function is like feeding the output of one machine into the other. In mathematical terms, the composite function $(f ext{ o } g)(x)$ means we first apply the function g to x, and then we apply the function f to the result. It's like a mathematical chain reaction! Understanding this concept of chaining functions is crucial for solving problems like the one we have. Think of it as a recipe: first you mix the ingredients (g(x)), then you bake the cake (f(x)). You can't bake the cake before mixing the ingredients, right? Similarly, we need to evaluate the inner function g(x) before we can evaluate the outer function f(x). This order of operations is key to understanding composite functions. So, with this fundamental understanding, we're well-equipped to tackle the problem head-on. Remember, composite functions are not just abstract mathematical concepts; they appear in many real-world applications, from computer programming to physics. The ability to work with composite functions unlocks a whole new level of problem-solving skills. So let's embrace the challenge and see how we can apply this knowledge to find the value of 'a' in our problem. This concept is like a building block for more advanced topics in calculus and analysis, so mastering it now will definitely pay off in the long run. Now that we've got a solid grasp of what composite functions are all about, let's dive into the specific problem at hand.

Problem Setup: Decoding the Given Information

Okay, let's break down the problem piece by piece. We're given two functions: f(x) = 2x + 3 and g(x) = x² - 3x + 5. Think of these as our mathematical tools. Function f is a simple linear function, while function g is a quadratic function. We're also told that (f o g)(a) = 21. This is the key piece of information! It tells us that when we plug 'a' into g, and then plug the result into f, we get 21. Our mission, should we choose to accept it, is to find the value of 'a' that makes this happen. This is like being a detective and piecing together clues to solve a mystery. The equation (f o g)(a) = 21 is our main clue, and the functions f(x) and g(x) are our tools for deciphering it. To really understand what this means, let's rewrite the composite function. Remember, (f o g)(a) is the same as f(g(a)). This notation is super helpful because it visually shows us the order of operations: we first find g(a), and then we use that result as the input for f. So, the problem is essentially asking us: what value(s) of 'a' will make f(g(a)) equal to 21? To answer this, we need to first figure out what g(a) is in terms of 'a'. We'll then plug that expression into f(x), and finally, we'll have an equation we can solve for 'a'. This might sound like a lot of steps, but don't worry, we'll take it slow and steady. Each step is a logical progression, and by the end, you'll see how it all fits together. The power of mathematics lies in its ability to break down complex problems into smaller, manageable steps. So, with our detective hats on, let's move on to the next step: finding g(a). We're well on our way to cracking this case!

Step-by-Step Solution: Unraveling the Mystery

Let's get our hands dirty with the math now! First, we need to find g(a). Remember, g(x) = x² - 3x + 5. So, to find g(a), we simply replace x with a: g(a) = a² - 3a + 5. Easy peasy, right? Now we have an expression for g(a). This is like finding the first piece of the puzzle. Next, we need to plug this expression into f(x). We know f(x) = 2x + 3. So, to find f(g(a)), we replace x in f(x) with our expression for g(a): f(g(a)) = 2(a² - 3a + 5) + 3. See how we're building upon each step? We started with g(a), and now we have f(g(a)). Now, let's simplify this expression. Distribute the 2: f(g(a)) = 2a² - 6a + 10 + 3. Combine the constants: f(g(a)) = 2a² - 6a + 13. We're getting closer! Remember, we know that f(g(a)) = 21. So, we can set our expression equal to 21: 2a² - 6a + 13 = 21. Now we have a quadratic equation! This is the home stretch. To solve for 'a', we need to rearrange the equation so it's in the standard quadratic form (ax² + bx + c = 0). Subtract 21 from both sides: 2a² - 6a - 8 = 0. Now, let's simplify by dividing the entire equation by 2: a² - 3a - 4 = 0. We've got a nice, clean quadratic equation to solve. There are a few ways to solve this: factoring, the quadratic formula, or completing the square. In this case, factoring looks like the easiest option. Can you think of two numbers that multiply to -4 and add up to -3? Those numbers are -4 and 1. So, we can factor the quadratic as: (a - 4)(a + 1) = 0. Now, we use the zero product property, which says that if the product of two factors is zero, then at least one of the factors must be zero. So, either a - 4 = 0 or a + 1 = 0. Solving these simple equations, we get: a = 4 or a = -1. And there you have it! We've found the possible values of 'a'. It's like finding the hidden treasure at the end of a map!

Verification and Conclusion: The Final Check

Before we declare victory, it's always a good idea to check our answers. This is like double-checking your work to make sure you haven't made any mistakes. We found two possible values for 'a': 4 and -1. Let's plug each one back into the original equation, f(g(a)) = 21, to see if they work. First, let's check a = 4: g(4) = 4² - 3(4) + 5 = 16 - 12 + 5 = 9 f(g(4)) = f(9) = 2(9) + 3 = 18 + 3 = 21 Great! a = 4 works. Now, let's check a = -1: g(-1) = (-1)² - 3(-1) + 5 = 1 + 3 + 5 = 9 f(g(-1)) = f(9) = 2(9) + 3 = 18 + 3 = 21 Awesome! a = -1 also works. Both values satisfy the original equation. So, we can confidently say that the solutions are a = 4 and a = -1. We've successfully solved the problem! We started with a composite function, broke it down step by step, and found the values of 'a' that make the equation true. This problem demonstrates the power of breaking down complex problems into smaller, manageable steps. By understanding the concept of composite functions, carefully substituting and simplifying, and using our algebraic skills, we were able to unravel the mystery and find the solutions. Math problems like this might seem daunting at first, but with practice and a systematic approach, you can conquer them! So, keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is full of fascinating puzzles just waiting to be solved. And who knows, maybe the next one you solve will unlock a whole new level of understanding!

In conclusion, by meticulously applying the definition of composite functions and utilizing algebraic techniques such as substitution, simplification, and factoring, we successfully determined the values of 'a' to be 4 and -1. This exercise not only reinforces our understanding of composite functions but also highlights the importance of methodical problem-solving in mathematics. The ability to break down complex problems into smaller, more manageable steps is a valuable skill that extends beyond the realm of mathematics and into various aspects of life. So, remember to embrace challenges, approach them with a systematic mindset, and never shy away from the rewarding journey of mathematical exploration.