Finding The Zero Of A Function: F(x) = 3√(x+3) - 6

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Hey guys! Today, we're diving into the exciting world of functions, specifically focusing on how to find the zero of a function. Now, what exactly is a zero of a function? Simply put, it's the value of x that makes the function f(x) equal to zero. In other words, it's where the graph of the function crosses the x-axis. We're going to tackle the function f(x) = 3√(x+3) - 6 and figure out its zero. So, grab your thinking caps, and let's get started!

Understanding the Zero of a Function

Before we jump into the nitty-gritty of solving this particular equation, let's make sure we're all on the same page about what we're trying to find. The zero of a function, sometimes also called the root or the x-intercept, is the value (or values) of x that satisfy the equation f(x) = 0. Think of it as the input value that causes the function to output zero. Understanding this fundamental concept is crucial, guys, because finding zeros is a common task in algebra, calculus, and many other areas of math and science. It helps us understand the behavior of the function, locate key points on its graph, and solve related problems. The zero of a function is a fundamental concept in mathematics. To be more specific, the zero of a function f(x) is a value x such that f(x) = 0. Geometrically, the zeros of a function are the points where the graph of the function intersects the x-axis. Finding the zeros of a function is a crucial step in understanding its behavior and solving related problems. For example, in physics, the zeros of a function might represent the equilibrium points of a system. In engineering, they might represent the points where a structure is stable. The process of finding the zeros of a function often involves algebraic manipulation and equation-solving techniques. Depending on the complexity of the function, different methods may be required, such as factoring, using the quadratic formula, or employing numerical methods. Let's proceed to solve the equation f(x) = 3√(x+3) - 6 = 0 step by step. First, we need to isolate the square root term. Then, we'll square both sides of the equation to eliminate the square root. Finally, we'll solve for x and verify our solution. So, keep that in mind as we move forward! We will use algebraic techniques to isolate the square root, eliminate it, and then solve the resulting equation.

Solving for the Zero of f(x) = 3√(x+3) - 6

Okay, now let's get our hands dirty and actually solve for the zero of our function, f(x) = 3√(x+3) - 6. Here's how we'll do it, step-by-step, making sure it’s super clear for everyone:

  1. Set the function equal to zero: The first thing we need to do is set our function, f(x), equal to zero. This is because we're looking for the value of x that makes the function output zero. So, we write: 3√(x+3) - 6 = 0

  2. Isolate the square root term: Next, we want to get the square root term by itself on one side of the equation. To do this, we'll add 6 to both sides: 3√(x+3) = 6

  3. Divide to simplify: Now, we'll divide both sides by 3 to further isolate the square root: √(x+3) = 2

  4. Square both sides: To get rid of the square root, we'll square both sides of the equation. Remember, whatever you do to one side, you gotta do to the other! (√(x+3))^2 = 2^2 This simplifies to: x + 3 = 4

  5. Solve for x: Finally, we solve for x by subtracting 3 from both sides: x = 4 - 3 x = 1

So, it looks like we've found a potential zero: x = 1. But, hold on a second! We're not quite done yet. Whenever we square both sides of an equation, especially when dealing with square roots, we need to check our answer to make sure it's not an extraneous solution. An extraneous solution is a value that we get when solving the equation, but it doesn't actually work when we plug it back into the original equation. This is super important, guys, because we don't want to accidentally include a wrong answer! Verifying our solution ensures that the value we found is indeed a zero of the function and not an extraneous solution introduced during the solving process. Extraneous solutions can arise when squaring both sides of an equation because this operation can introduce solutions that do not satisfy the original equation. Therefore, checking the solution in the original equation is a necessary step to ensure its validity. To check our solution, we substitute x = 1 back into the original equation: f(x) = 3√(x+3) - 6. Let's do it!

Verification of the Solution

Alright, we've arrived at a potential solution, x = 1, but as we discussed, we need to verify it to make sure it's the real deal. This is a crucial step, especially when dealing with square roots, to avoid those sneaky extraneous solutions. So, let's plug x = 1 back into our original function, f(x) = 3√(x+3) - 6, and see what happens:

  1. Substitute x = 1: Replace x with 1 in the equation: f(1) = 3√(1+3) - 6

  2. Simplify: Now, let's simplify step-by-step: f(1) = 3√4 - 6 f(1) = 3 * 2 - 6 f(1) = 6 - 6 f(1) = 0

Woohoo! We got f(1) = 0. This means that when we plug x = 1 into our function, the output is indeed zero. So, our solution checks out! This confirms that x = 1 is a valid zero of the function f(x) = 3√(x+3) - 6. We've successfully navigated the algebra and verification steps to find the zero. This process of verification is not just a formality; it's a fundamental practice in mathematics to ensure the correctness of solutions, especially when dealing with operations that can introduce extraneous roots. The verification step provides us with confidence that our solution is accurate and that x = 1 is indeed the value where the function crosses the x-axis. Moreover, understanding this process helps in building a deeper conceptual understanding of functions and their zeros. By verifying our solution, we ensure that it is consistent with the original equation and that no algebraic errors were made during the solving process. The verification step is a necessary part of problem-solving, and it's a good habit to develop for any mathematical problem, not just those involving square roots. It can help you catch mistakes and ensure that your final answer is correct. Therefore, it reinforces the idea that mathematical problem-solving is a careful and rigorous process.

Conclusion

Awesome job, guys! We successfully found the zero of the function f(x) = 3√(x+3) - 6. By setting the function equal to zero, isolating the square root, squaring both sides, and solving for x, we arrived at the potential solution x = 1. And, importantly, we verified our solution by plugging it back into the original equation and confirming that it indeed makes the function equal to zero. This whole process highlights the importance of not just solving, but also verifying our solutions, especially when dealing with square roots and other operations that can introduce extraneous solutions. Finding the zero of a function is a key concept in mathematics with wide-ranging applications in various fields, from physics to engineering. Understanding how to approach these problems systematically, including the verification step, will serve you well in your mathematical journey. So, keep practicing, keep exploring, and keep having fun with math! Remember, the zero of a function is where the magic happens, where the function crosses the x-axis, and where we can unlock valuable insights about its behavior. You've now got another tool in your mathematical toolkit, and you're one step closer to mastering the world of functions. Keep up the great work!