Solving Composite Functions: Finding 'x' When (g ∘ H ∘ F)(x) = 11
Hey guys! Let's dive into a cool math problem involving composite functions. We're given three functions: f(x), g(x), and h(x), and our goal is to find the value(s) of 'x' that satisfy a specific composite function equation. Sounds like fun, right? Don't worry, we'll break it down step by step, making it super easy to understand. This is a classic problem in algebra, and understanding how to solve it is a fundamental skill. So, let's get started and unlock the secrets of composite functions!
Understanding the Problem: The Setup
Okay, so here's what we're working with. We have three functions defined as follows:
- f(x) = x + 2
- g(x) = -3 - 2x
- h(x) = x² + 3x - 4
Our main task is to find the value(s) of 'x' when the composite function (g ∘ h ∘ f)(x) equals 11. The notation (g ∘ h ∘ f)(x) means we're applying the functions in a specific order: first f, then h, and finally g. It's like a mathematical assembly line! To solve this, we'll start from the inside out, meaning we'll first apply f(x) to x, then use that result as the input for h(x), and finally use that result as the input for g(x). Understanding this order of operations is absolutely crucial. Let's remember what each function does. The function f(x) simply adds 2 to the input x. The function g(x) multiplies the input by -2 and subtracts 3. The function h(x) squares the input, adds three times the input and subtracts 4. Keeping these in mind, we can start breaking down our composite function and solve for x!
Breaking Down the Composite Function: Step-by-Step Solution
Alright, time to get our hands dirty and actually solve this thing! We'll start by finding the expression for (g ∘ h ∘ f)(x). Remember, we work from the inside out. First, let's find h(f(x)).
Since f(x) = x + 2, we substitute (x + 2) into h(x):
h(f(x)) = h(x + 2) = (x + 2)² + 3(x + 2) - 4
Now, let's simplify this:
h(f(x)) = (x² + 4x + 4) + (3x + 6) - 4 h(f(x)) = x² + 7x + 6
Great! We've found h(f(x)). Now, we need to find g(h(f(x))). We'll substitute the expression we just found (x² + 7x + 6) into g(x):
g(h(f(x))) = g(x² + 7x + 6) = -3 - 2(x² + 7x + 6)
Let's simplify this:
g(h(f(x))) = -3 - 2x² - 14x - 12 g(h(f(x))) = -2x² - 14x - 15
So, we've found that (g ∘ h ∘ f)(x) = -2x² - 14x - 15. The problem states that this composite function equals 11. Therefore, we can set up the equation:
-2x² - 14x - 15 = 11
Let's continue our quest to conquer this mathematical puzzle, now that we've unpacked the beast, let's tame it!
Solving the Quadratic Equation: Finding the Values of 'x'
Okay, we've reached a quadratic equation. To solve it, we need to set the equation to zero. Let's subtract 11 from both sides:
-2x² - 14x - 15 - 11 = 0 -2x² - 14x - 26 = 0
Now, we can simplify this by dividing the entire equation by -2:
x² + 7x + 13 = 0
To solve for x in a quadratic equation, you have a couple of options: factoring, completing the square, or using the quadratic formula. In this case, factoring might not be the easiest route. Let's use the quadratic formula, which is a reliable method for any quadratic equation.
The quadratic formula is: x = (-b ± √(b² - 4ac)) / 2a
In our equation, x² + 7x + 13 = 0, we have:
a = 1 b = 7 c = 13
Now, let's plug these values into the quadratic formula:
x = (-7 ± √(7² - 4 * 1 * 13)) / (2 * 1) x = (-7 ± √(49 - 52)) / 2 x = (-7 ± √(-3)) / 2
Oops! We have a negative number under the square root. This means we'll get complex roots, not real roots. Since the problem asks for real values of x, it seems there might be a mistake in our calculations or in the original problem statement, or possibly in the answer choices. Looking back at the problem, we can see there was an error in our simplification, therefore the equation should have been: x² + 7x + 6 = 0 after evaluating h(f(x)). So when plugging that in, we have:
g(x² + 7x + 6) = -3 - 2(x² + 7x + 6)
g(x² + 7x + 6) = -3 - 2x² - 14x - 12
g(x² + 7x + 6) = -2x² - 14x - 15
Then, we set this equal to 11, -2x² - 14x - 15 = 11, and then we get -2x² - 14x - 26 = 0. Divide by -2 and we get x² + 7x + 13 = 0. So, the math checks out, but we still end up with complex roots. It is important to note that while complex solutions are mathematically valid, they are not always the focus of introductory algebra problems. Therefore, let's revisit our simplification, or we should consider if the answer choices are possibly flawed. In conclusion, the original question is leading to a problem that has complex roots and should be changed to have real root solutions.
Analyzing the Answer Choices and Conclusion
Based on our calculations, we've found complex roots, which don't match the typical answer choices provided. In the original, the choices include real numbers, which suggests there may be a calculation error in the problem or the answer choices. Usually, these problems are designed to have neat, easy-to-find solutions, not complex ones. Since we can't provide a real number answer matching the provided choices, a re-evaluation of the process will be performed.
Therefore, let's re-evaluate our calculations to check where the mistake might have been. After evaluating h(f(x)) = x² + 7x + 6. We found g(x² + 7x + 6) = -2x² - 14x - 15. After we set that to equal 11, we ended up with an answer that did not match the choices. Hence, it can be derived that the original problem is most likely flawed and would need to be corrected to produce real numbers. So in conclusion, we tried our best to derive a proper solution, but we ended up with complex roots, and therefore, we cannot find a valid solution to the problem with the information given. To ensure we can come to a valid answer, the question will need to be revised.
So, the answer choices provided are not correct based on the given functions. It's super important to double-check your work, especially when dealing with composite functions and quadratic equations. Always review your steps and make sure you didn't make any arithmetic errors. If you're ever unsure, don't hesitate to ask for help!
