Calculating F(-7): Step-by-Step Guide

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Hey math enthusiasts! Today, we're diving into a straightforward but crucial concept in algebra: evaluating a function. Specifically, we'll learn how to find the value of a function, denoted as f(x), at a particular point. In our case, we'll be tackling the problem of finding f(-7) given the function f(x) = -2x² - 4x + 13. This process, also known as function evaluation, is a fundamental skill in mathematics, forming the basis for more advanced topics like calculus and differential equations. So, let's get started and break down this problem in an easy-to-understand way!

Understanding Function Evaluation

Function evaluation is essentially substituting a specific value for the variable (in this case, 'x') in the function's expression and then simplifying the result. Think of a function like a machine; you input a value, and the machine processes it according to a predefined set of rules (the function's formula), ultimately spitting out a unique output. In the given function, f(x) = -2x² - 4x + 13, the function tells us what to do with the input 'x'. We take 'x', square it, multiply it by -2, multiply 'x' by -4, and then add 13. To find f(-7), we just have to replace every instance of 'x' in the expression with -7. This is the core concept of function evaluation, a skill you'll use constantly as you progress in mathematics. This operation is not just limited to algebra; you can also see it in different programming languages too. For example, in Python you will define a function like def f(x): return -2*x**2 - 4*x + 13. This function can also be evaluated with the f(-7) command.

The Steps

Let's break down the process step by step to make it crystal clear:

  1. Substitution: Replace every 'x' in the function with '-7'. Our function becomes: f(-7) = -2(-7)² - 4(-7) + 13.
  2. Exponents: Calculate the exponent first. Remember, according to the order of operations (PEMDAS/BODMAS), exponents come before multiplication and addition/subtraction. So, (-7)² = 49. The function now looks like this: f(-7) = -2(49) - 4(-7) + 13.
  3. Multiplication: Perform the multiplications. We have -2 * 49 = -98 and -4 * -7 = 28. The function simplifies to: f(-7) = -98 + 28 + 13.
  4. Addition/Subtraction: Finally, perform the addition and subtraction from left to right: -98 + 28 = -70, and -70 + 13 = -57. Therefore, f(-7) = -57. And just like that, we have found our answer!

Detailed Calculation

Okay, guys, let's take a closer look at the step-by-step calculation to ensure everything is super clear. Remember, we are given the function: f(x) = -2x² - 4x + 13 and we need to find f(-7).

  1. Substitution: Replace 'x' with -7: f(-7) = -2(-7)² - 4(-7) + 13. This step is about replacing the variable with the given value. It's crucial to make sure you substitute the value in every place where 'x' appears.
  2. Exponents: Calculate (-7)². Remember, (-7)² = (-7) * (-7) = 49. Our equation now looks like this: f(-7) = -2(49) - 4(-7) + 13. Note that the negative sign inside the parenthesis disappears since a negative number multiplied by a negative number becomes positive.
  3. Multiplication: Multiply -2 by 49 and -4 by -7: f(-7) = -98 + 28 + 13. Pay close attention to the signs here; a negative times a positive is negative, and a negative times a negative is positive.
  4. Addition/Subtraction: Perform the addition and subtraction: -98 + 28 = -70. Then, -70 + 13 = -57. So, we arrive at the final answer: f(-7) = -57. Voila! We've successfully calculated the value of the function at x = -7. Practicing more examples will improve your ability to solve this type of problem.

Order of Operations Reminder

It's important to remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) or BODMAS (Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)). This order dictates the sequence in which you perform operations to ensure you get the correct answer. In our problem, we first dealt with the exponent, then the multiplications, and finally the additions and subtractions. Getting the order wrong will lead you to the wrong answer. Make sure to keep this order in mind when solving mathematical problems!

Why Function Evaluation Matters

So, why is understanding how to evaluate functions so important, you might ask? Well, it's not just about solving this particular problem; it's about building a strong foundation in algebra and beyond. Function evaluation is a gateway to several core mathematical concepts: understanding relationships between variables, modeling real-world phenomena, and grasping the fundamentals of calculus. Whether you're interested in science, engineering, economics, or computer science, function evaluation is a foundational skill. It lets you analyze patterns, make predictions, and understand how changes in one variable impact others. Therefore, mastering this skill is an investment in your future mathematical understanding.

Real-World Applications

Function evaluation isn't just an abstract concept; it has tons of real-world applications. For instance, in physics, you might use a function to calculate the position of a moving object over time, given its initial velocity and acceleration. In finance, you could use a function to model the growth of an investment, taking into account interest rates and time. Even in everyday life, you implicitly use functions when you calculate things like the cost of groceries or the distance you need to travel to get somewhere. The functions can also be used in more complex situations, like the evaluation of machine learning models. So, keep practicing, keep learning, and remember that these skills are valuable! They are fundamental building blocks for many problems.

Tips for Success

To become a pro at evaluating functions, here are a few tips and tricks:

  1. Practice: The more problems you solve, the more comfortable you'll become. Practice makes perfect! Work through a variety of examples, and don't be afraid to make mistakes—it's part of the learning process. The more you do, the faster you will become at these problems. Start with easy examples and then increase the difficulty.
  2. Order of Operations: Always, always, always remember PEMDAS/BODMAS. It's the key to getting the right answer. Writing down the intermediate steps will help you.
  3. Pay Attention to Signs: Be super careful with positive and negative signs. A small mistake here can completely change your answer. Ensure you check everything. It's very easy to miss a negative sign and get the wrong answer.
  4. Double-Check Your Work: After you've found the answer, go back and re-evaluate the function. Make sure you haven't missed any steps or made any calculation errors. This helps to catch any mistakes you may have made.
  5. Use Technology Wisely: While it's important to understand the process, don't hesitate to use a calculator or online tools to check your work. These tools can be especially useful for more complex calculations. Use these to check your answer and to make sure your work is right.

Conclusion

Alright, guys, that wraps up our guide on how to evaluate f(-7) for the function f(x) = -2x² - 4x + 13! We've covered the basics of function evaluation, walked through the steps in detail, and highlighted why this skill is so crucial. Remember to practice regularly, pay attention to the order of operations and signs, and always double-check your work. Keep up the great work, and happy calculating! With practice, you'll become a function evaluation master in no time. Function evaluation is an essential skill in mathematics and has many applications in everyday life. Good luck with your future math endeavors!