Solving For G(x) = 0 Given F(x) And G(x) Relationship

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Let's dive into this math problem together, guys! We've got a function f with some known values, and another function g defined in terms of f. Our mission, should we choose to accept it, is to figure out what makes g(x) equal to zero. It sounds like a fun puzzle, so let’s get started!

Understanding the Problem

Okay, so first things first, let’s break down what we know. We are given that f(3) = 0 and f(6) = -4. These are crucial pieces of information because they tell us something specific about the function f. Specifically, we know that when the input to f is 3, the output is 0. This is super important! We also know f(6) = -4, but that's slightly less directly helpful for finding when g(x) = 0, but it might help us to understand how the function f behaves generally. The other key piece of information is that g(x) is defined as f(x - 4)*. This means that to evaluate g at any value of x, we first subtract 4 from x, and then we plug that result into the function f. This is what we call a horizontal translation of the function f. Understanding this relationship between f and g is the key to solving the problem. The goal here is to find the value(s) of x that make g(x) equal to zero. This means we're looking for the input(s) to g that result in an output of 0. Let’s think about how the definition of g can help us find this.

The Core Concept: Finding the Zeros

The zeros of a function are the values of x that make the function equal to zero. In our case, we want to find the zeros of the function g. We know that g(x) = f(x - 4). So, g(x) will be zero whenever f(x - 4) is zero. This is a critical connection! We already know one value that makes f equal to zero: f(3) = 0. This information is going to be our starting point. We need to figure out what value of x will make the input to f equal to 3, because we know that f of that input will be zero. This is where the “x - 4” part of the definition of g(x) comes into play. By understanding how the input of f is linked to the input of g, we can effectively reverse-engineer the solution.

Solving for g(x) = 0

Here’s the heart of the problem-solving process. We know that g(x) = f(x - 4). We want to find the values of x for which g(x) = 0. This means we need to find the x values that satisfy the equation:

f(x - 4) = 0

Remember, we know that f(3) = 0. So, we need to make the input to f, which is (x - 4), equal to 3. This gives us a simple equation to solve:

x - 4 = 3

To solve for x, we just add 4 to both sides of the equation:

x = 3 + 4

x = 7

So, we’ve found a solution! When x is 7, g(x) will be equal to zero. Let’s double-check this to make sure it makes sense. If we plug x = 7 into g(x), we get:

g(7) = f(7 - 4) = f(3)

And we know that f(3) = 0, so:

g(7) = 0

Yep, it works! This confirms that x = 7 is indeed a solution to g(x) = 0. It’s always a good idea to double-check your work, especially in math problems, to make sure you haven’t made any silly mistakes. This is a great way to build confidence in your answer.

Why Other Values Might Not Work

Now, you might be wondering, why don't we use the information that f(6) = -4? Well, that information is helpful for understanding the general behavior of f, but it doesn't directly help us find where g(x) = 0. We needed to focus on where f itself is equal to zero. The fact that f(6) is -4 tells us that when the input to f is 6, the output is -4, which is not zero. So, this doesn't directly help us solve for g(x) = 0. Remember, we are specifically looking for the input values that make g equal to zero. Knowing where f takes on other values, while interesting, is not the direct path to our solution.

The Final Answer

Alright, guys, we've done it! We've successfully navigated through the problem and found the solution. Based on the given information, we determined that:

x = 7

is a solution to the equation g(x) = 0. This means that when x is 7, the function g outputs zero. This is our final answer, and we can be confident in it because we’ve carefully worked through each step, checked our work, and made sure our reasoning is sound. Solving problems like this is all about understanding the relationships between functions and using the given information strategically.

Key Takeaways

Let's recap the main things we learned from solving this problem:

  1. Understanding Function Composition: Recognizing that g(x) is a composite function, meaning it's a function within a function (f(x - 4)), is crucial. This allows us to break down the problem into smaller parts.
  2. Finding Zeros: The core concept is identifying the zeros of a function, which are the input values that make the function's output zero. In this case, we needed to find the zeros of g(x).
  3. Using Given Information: The information about f(3) = 0 was the key to unlocking the solution. We used this to determine when f(x - 4) would also be zero.
  4. Solving Equations: We had to set up and solve a simple equation (x - 4 = 3) to find the value of x that satisfies g(x) = 0.
  5. Checking Your Work: It’s always a good practice to double-check your solution to ensure it makes sense and that you haven’t made any calculation errors. This helps build confidence in your answer.

By mastering these concepts, you'll be well-equipped to tackle similar problems involving function composition and finding zeros. Keep practicing, and you'll become a pro at solving these types of mathematical puzzles!

Practice Problems

To really solidify your understanding, here are a couple of practice problems you can try:

  1. If f(2) = 0 and g(x) = f(x + 1), find the solution to g(x) = 0.
  2. If f(-1) = 0 and g(x) = f(x - 3), find the solution to g(x) = 0.

Try working through these problems using the same steps we used in the example. Remember to break down the problem, use the given information, and check your work. Good luck, and happy problem-solving!