Unraveling F(a+4): A Deep Dive Into Function Evaluation

Hey math enthusiasts! Let's dive into a fascinating function problem. We're given a function, f(x), defined by a set of rules. Our mission? To figure out the value of f(a+4), where 'a' is a negative number. Sounds intriguing, right? This problem is a fantastic blend of understanding function definitions, evaluating expressions, and recognizing conditions. Get ready to flex those math muscles! We'll break down the problem step-by-step, making sure we understand every twist and turn. By the end, you'll be a pro at tackling similar function problems. Let's get started!

Understanding the Function's Rules

First, let's get friendly with our function f(x). It's not just any function; it's a piecewise function. What's that, you ask? Well, it's like a function with multiple personalities. The rule it follows depends on the value of x. Here's the breakdown:

• If x is greater than or equal to 4 (x ≥ 4), then f(x) = x² + 3. This means that if we plug in a number for x that's 4 or bigger, we use the formula x² + 3 to find the output.
• If x is less than 4 (x < 4), then f(x) = -x + 4. This means that if we plug in a number for x that's smaller than 4, we use the formula -x + 4.

So, the key here is to figure out which rule applies based on the value of the input. Remember, each rule has its own conditions. Now, let's move on to the next step: determining the value of f(a+4).

To really grasp this, let's use an example. Imagine x = 5, since 5 >= 4, we would use the first rule: f(5) = 5² + 3 = 28. Now, if x = 2 (which is < 4), we would use the second rule: f(2) = -2 + 4 = 2. Pretty straightforward, right? This will be very helpful in understanding how to solve f(a+4).

Now, before we move on, I want to emphasize that it's crucial to pay close attention to the conditions for each rule. This will be the deciding factor in how we solve the problem. Let's keep this in mind as we approach f(a+4).

Now we've gone over the basic rules of the function, and it's time to test your knowledge of piecewise functions. This includes understanding the conditions that change the function's output. So, what is the value of f(a+4)? Keep reading, guys, we are getting closer to solving the equation!

Evaluating f(a+4): The Core of the Problem

Alright, now for the main event: finding the value of f(a+4). Remember, 'a' is a negative number. Here's where things get interesting. We need to figure out which rule of our function applies when we plug in a+4.

Since 'a' is negative, let's think about a+4. Because 'a' is negative, a is a number less than zero. So, when we add 4 to 'a', we want to see if the resulting value is greater than or equal to 4, or less than 4.

Here's how we can think about this:

• If a+4 ≥ 4, then we'll use the rule f(x) = x² + 3.
• If a+4 < 4, then we'll use the rule f(x) = -x + 4.

Since 'a' is a negative number, 'a' must be less than 0. Let's see what happens to a+4. If a = -1, then a+4 = -1 + 4 = 3. If a = -5, then a+4 = -5 + 4 = -1. Notice how in both cases, the result is less than 4?

So, if a is any negative number, a+4 will always be less than 4. So, we'll use the rule f(x) = -x + 4. But remember, we're not plugging in x, we're plugging in a+4. Thus, f(a+4) = -(a+4) + 4.

Now let's simplify that expression:

• f(a+4) = -(a+4) + 4*.
• f(a+4) = -a - 4 + 4*.
• f(a+4) = -a*.

Therefore, the value of f(a+4) when 'a' is a negative number is -a. This means that the correct option is B. -a.

Deep Dive: Why Other Options Are Incorrect

Let's analyze why the other options aren't the right answer. Understanding why incorrect answers are wrong is just as important as knowing the correct one. It reinforces our understanding and helps prevent future mistakes.

• Option A: a² + 8a + 16. This option represents the result of (a+4)². This suggests that we might have incorrectly applied the rule x² + 3, which we would have used if a+4 ≥ 4. However, since 'a' is negative, this isn't the correct rule.
• Option C: a² + 19. This seems to be a result of applying the incorrect rule. There's no direct path within our function to get this result. It is not clear how this solution could be obtained based on our rules, and given the context, we can assume that this option is completely wrong.
• Option D: a² + 8a + 13. This option is similar to option A, and also incorrect. This option might be a misunderstanding of the function's rules or an error in calculation, since 'a' is a negative number, and we need to use the rule of f(x) = -x + 4.

By carefully examining each option, we can confirm our correct answer and understand the common pitfalls in this type of problem. It's all about paying attention to the details and carefully following the rules of the function!

Key Takeaways and Final Thoughts

Congratulations, guys! You've successfully navigated the world of piecewise functions and found the value of f(a+4). Here's what we learned:

1. Understanding Piecewise Functions: Piecewise functions change their behavior based on the input value. We need to carefully identify which rule applies.
2. Evaluating Expressions: Substituting a+4 into the correct rule is essential. We use the rule that corresponds to the condition met by a+4.
3. The Importance of 'a': The fact that 'a' is negative is key. It helps us determine which rule to use. Without this information, we could not solve the problem properly.

Remember, practice makes perfect! Try similar problems, experiment with different inputs, and always double-check your work. Keep up the amazing work! You are now well-equipped to tackle similar function problems. Keep practicing and keep that math spark alive!

This problem showed us the beauty of functions and how they adapt based on conditions. The most important thing is to focus on each step, pay attention to every detail, and not be afraid to break down complicated problems into simpler ones. You've got this!