Find F(-4) For F(x) = X^2 - 5x - 6: Step-by-Step Solution

Have you ever been presented with a function and asked to find its value at a specific point? It's a fundamental concept in algebra and calculus, and it's easier than it might seem! In this article, we'll break down a common type of problem: evaluating a quadratic function. Specifically, we'll walk through finding the value of f(-4) for the function f(x) = x^2 - 5x - 6. This is a classic example that helps illustrate the process of function evaluation, and by understanding this, you'll be well-equipped to tackle similar problems. So, let’s dive in and make math a little less mysterious and a lot more fun!

Understanding Function Notation

Before we jump into the solution, let's make sure we're all on the same page with function notation. The expression f(x) simply means “the value of the function f at x”. Think of a function as a machine: you put in a value (x), and the machine performs some operations on it and spits out a result (f(x)). The key here is that the function defines a relationship between inputs and outputs. When we write f(-4), we're asking: “What is the output of the function f when the input is -4?” This notation is incredibly powerful because it allows us to represent complex relationships in a concise way. Understanding this notation is crucial not just for solving problems like this one, but also for grasping more advanced concepts in mathematics and other fields. It's the basic language of functions, and mastering it opens doors to a whole new world of mathematical understanding. So, remember, f(x) is your friendly guide to navigating the world of functions!

Step-by-Step Solution: Evaluating f(-4)

Now, let's get down to business and find f(-4) for the function f(x) = x^2 - 5x - 6. The process is straightforward: we simply substitute -4 for every x in the function's expression. This is the core of function evaluation – replacing the variable with the specific value we're interested in. So, wherever you see an x, mentally replace it with -4, being careful to maintain the correct order of operations. This might seem simple, but it's a fundamental skill in mathematics, and mastering it will make solving more complex problems much easier. Attention to detail is key here; make sure you're substituting correctly and handling the negative signs properly. Let’s get started with our substitution and see how the function transforms as we plug in -4. This first step sets the stage for the rest of the solution, so let’s make it count!

Step 1: Substitute -4 for x

Our function is f(x) = x^2 - 5x - 6. To find f(-4), we replace each x with -4: f(-4) = (-4)^2 - 5(-4) - 6. It's crucial to use parentheses around -4 to ensure we handle the negative sign correctly, especially when dealing with exponents. This simple act of using parentheses can be the difference between a correct answer and a frustrating mistake. Think of it as a protective shield, guarding the sign of your number and ensuring it participates properly in the mathematical operations. Now that we've made the substitution, the next step is to simplify the expression, following the order of operations. This means we'll tackle the exponent first, then multiplication, and finally, addition and subtraction. So, let’s move on to the next step and see how this expression simplifies!

Step 2: Simplify the Expression

Now, let's simplify the expression we obtained after substitution: f(-4) = (-4)^2 - 5(-4) - 6. Following the order of operations (PEMDAS/BODMAS), we first evaluate the exponent: (-4)^2 = 16. Remember that a negative number squared becomes positive. This is a common area for errors, so always double-check your signs. Next, we perform the multiplication: -5(-4) = 20. Again, a negative times a negative gives a positive. Now our expression looks like this: f(-4) = 16 + 20 - 6. All that's left is addition and subtraction, which we perform from left to right. So, 16 + 20 = 36, and then 36 - 6 = 30. Therefore, f(-4) = 30. We've successfully simplified the expression and found the value of the function at x = -4. This step-by-step simplification is the heart of the solution, and it highlights the importance of following the order of operations to arrive at the correct answer.

Step 3: State the Final Answer

After carefully substituting and simplifying, we've arrived at our final answer: f(-4) = 30. This means that when we input -4 into the function f(x) = x^2 - 5x - 6, the output is 30. It's always a good idea to double-check your work, especially in math problems. Go back through each step and make sure you haven't made any arithmetic errors or sign mistakes. Once you're confident in your answer, you can clearly state it. In this case, the final answer is 30, which corresponds to option D from the original problem. Stating the answer clearly and confidently is the final touch to a well-solved problem. It demonstrates that you not only understand the process but can also communicate your results effectively.

Common Mistakes to Avoid

When evaluating functions, especially quadratic functions, there are a few common pitfalls that students often encounter. Being aware of these mistakes can help you avoid them and ensure you get the correct answer. One of the most frequent errors is mishandling negative signs. Remember that squaring a negative number results in a positive number, and multiplying two negative numbers also yields a positive result. Always double-check your signs to avoid this error. Another common mistake is not following the order of operations (PEMDAS/BODMAS). Make sure you perform exponents before multiplication and division, and multiplication and division before addition and subtraction. Skipping steps or performing operations in the wrong order can lead to incorrect results. Finally, careless arithmetic errors can derail even the most well-intentioned efforts. Take your time, write neatly, and double-check your calculations to minimize these mistakes. By being mindful of these common errors, you can significantly improve your accuracy and confidence in solving function evaluation problems.

Practice Makes Perfect

Now that we've walked through the solution step-by-step, the best way to solidify your understanding is to practice! Try evaluating the function f(x) = x^2 - 5x - 6 for other values of x, such as 0, 1, or even other negative numbers. This will help you become more comfortable with the substitution and simplification process. You can also find similar problems in your textbook or online and work through them. The more you practice, the more natural this process will become, and the more confident you'll feel in your ability to evaluate functions. Think of it like learning a musical instrument or a new language – consistent practice is the key to mastery. So, grab a pencil and paper, find some practice problems, and start honing your function evaluation skills!

In conclusion, evaluating a function at a given point is a fundamental skill in mathematics. By understanding function notation, following the order of operations, and avoiding common mistakes, you can confidently solve these types of problems. Remember to practice regularly to reinforce your understanding and build your skills. We successfully found that f(-4) = 30 for the function f(x) = x^2 - 5x - 6. Keep practicing, and you'll be a function evaluation pro in no time!

For more in-depth explanations and examples of functions, check out Khan Academy's Functions and Equations section. It's a fantastic resource for learning and reinforcing mathematical concepts.