Evaluating F(2) For F(x) = 2x + 3: A Simple Guide

by Alex Johnson 50 views

Let's dive into a straightforward math problem: evaluating a function at a specific point. In this case, we want to find the value of f(2), given that f(x) = 2x + 3. This is a foundational concept in algebra and is crucial for understanding more complex mathematical ideas. We'll break down the process step by step, making it easy to grasp, even if you're just starting your mathematical journey. Understanding function evaluation is like learning a new language; once you understand the grammar, you can express a multitude of ideas. This article provides a comprehensive guide to understanding the function and how to use it. By following these steps you can easily solve function questions.

Understanding the Function

Before we jump into evaluating f(2), let's make sure we understand what the function f(x) = 2x + 3 represents. A function is like a machine: you input a value (in this case, x), and the machine performs some operations on it and spits out a result. In our function, whatever value we put in for x, it will first be multiplied by 2, and then 3 will be added to the result. Think of x as a placeholder; it can be any number, and the function tells us what to do with it. This is a linear function, which means when graphed, it will form a straight line. The 2x part determines the slope of the line, and the +3 part determines where the line intersects the y-axis. Understanding these basics allows us to tackle more complex problems involving function transformations, compositions, and inverses. Now, consider some different values for x. If x = 0, f(0) would be 2*(0) + 3 = 3. If x = 1, f(1) would be 2*(1) + 3 = 5. Each input results in a specific output, forming a coordinate point on the line. Recognizing the structure of the function is the first step towards mastering it. This basic understanding is also fundamental to more advanced topics such as calculus, where the behavior of functions is analyzed in detail.

The Substitution Step

Now that we have a handle on what the function means, let's get to the heart of the problem: evaluating f(2). This means we're going to replace every instance of x in the function f(x) = 2x + 3 with the number 2. So, wherever we see x, we'll put a 2 in its place. This might seem simple, but it's a critical step. Many errors in math come from not carefully substituting values correctly. It's like following a recipe: if you misread an ingredient amount, the final dish won't turn out as expected. So, let's be meticulous. We rewrite f(x) = 2x + 3 as f(2) = 2(2) + 3. Notice how x has been replaced with 2. This is the essence of function evaluation. This substitution technique is applicable to all types of functions, whether they are polynomial, trigonometric, or exponential. The key is to identify the variable and replace it with the given value. Correct substitution is the foundation for obtaining the correct answer. Consider substituting other values, such as -1 or 0.5, to reinforce your understanding. The process remains the same, regardless of the input value. Accurate substitution ensures that the subsequent calculations will lead to the correct evaluation of the function.

Performing the Calculation

After the substitution, we now have f(2) = 2(2) + 3. The next step is to perform the calculation according to the order of operations (often remembered by the acronym PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction). In our expression, we first need to perform the multiplication: 2(2). This equals 4. So now our expression looks like this: f(2) = 4 + 3. The final step is to perform the addition: 4 + 3. This equals 7. Therefore, f(2) = 7. This means that when we input 2 into our function f(x) = 2x + 3, the output is 7. Always remember to follow the order of operations to avoid mistakes. Even in simple calculations, adherence to this rule is crucial. For instance, if we had addition before multiplication, we would get a completely different answer. Practice with similar expressions to build confidence and accuracy in your calculations. This skill will be invaluable as you progress to more complex mathematical problems. Reviewing the order of operations regularly can prevent common errors and enhance your problem-solving abilities.

The Result

So, after all our hard work, we've arrived at the answer: f(2) = 7. This is the value of the function f(x) = 2x + 3 when x is equal to 2. We've successfully evaluated the function at a specific point. This might seem like a small accomplishment, but it's a fundamental building block in mathematics. It allows us to understand how functions behave and to make predictions based on them. The result is not just a number; it's a piece of information about the function's behavior. It tells us that at x = 2, the function's output is 7, which can be plotted as a point (2, 7) on the graph of the function. This understanding is vital for applications in science, engineering, and economics, where functions are used to model real-world phenomena. Keep practicing with different functions and different input values to solidify your understanding and build your mathematical skills. Function evaluation is a powerful tool, and mastering it will open doors to more advanced mathematical concepts.

Practice Problems

To solidify your understanding, let's work through a couple of practice problems. These will help you apply the concepts we've discussed and build confidence in your ability to evaluate functions.

Practice Problem 1

Evaluate g(x) = -3x + 5 at x = 4.

Solution:

  1. Substitute x with 4: g(4) = -3(4) + 5
  2. Perform the multiplication: g(4) = -12 + 5
  3. Perform the addition: g(4) = -7

Therefore, g(4) = -7.

Practice Problem 2

Evaluate h(x) = x/2 - 1 at x = 6.

Solution:

  1. Substitute x with 6: h(6) = 6/2 - 1
  2. Perform the division: h(6) = 3 - 1
  3. Perform the subtraction: h(6) = 2

Therefore, h(6) = 2. These practice problems demonstrate the consistent process of substitution and calculation required for function evaluation. By working through these examples, you can reinforce your understanding and improve your accuracy. Continue practicing with various functions and values to develop a strong foundation in this essential mathematical skill.

Conclusion

In conclusion, we've walked through the process of evaluating f(2) for the function f(x) = 2x + 3. We started by understanding the function, then carefully substituted the value of x, performed the calculation, and arrived at the result: f(2) = 7. This simple example illustrates a fundamental concept in mathematics. By mastering function evaluation, you're building a strong foundation for more advanced topics. Practice regularly and don't hesitate to tackle more complex functions. The key is to break down the problem into smaller steps and to be meticulous in your calculations. Understanding and applying these basic mathematical operations can help you to solve simple and complex problems. The ability to confidently evaluate functions is a valuable skill in many fields, from science and engineering to economics and finance. Keep exploring and expanding your mathematical knowledge!

For further learning about functions, check out Khan Academy's Functions and Equations Section.