Evaluating F(36) For F(x) = (2/3)x - 12: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem where we need to evaluate a function. Specifically, we want to figure out what f(36) is when we know that f(x) = (2/3)x - 12. Don't worry; it's not as scary as it looks! We'll break it down step-by-step so it's super easy to follow. Understanding function evaluation is a crucial skill in mathematics, as functions are the building blocks for more advanced concepts in algebra, calculus, and beyond. This particular problem involves a linear function, which is the simplest type of function, making it a great starting point for mastering function evaluation. Let's get started and see how to solve this problem together!

Understanding the Function

Before we jump into plugging in numbers, let's make sure we understand what the function f(x) = (2/3)x - 12 is telling us. In mathematical terms, a function is like a little machine: you put a number in (that's our x), and it spits out another number based on a specific rule. This rule defines how the input x is transformed into the output f(x). For a stronger grasp, consider exploring different types of functions and their properties. Understanding the fundamental characteristics of functions, such as linearity, is key to success in algebra and calculus. Linear functions, like the one we're working with, have the form f(x) = mx + b, where m represents the slope and b represents the y-intercept. In our case, m = 2/3 and b = -12. This means that for every increase of 1 in x, the value of f(x) increases by 2/3, and the line crosses the y-axis at -12. Visualizing this on a graph can help solidify your understanding of linear functions and how they behave. Furthermore, functions are not limited to simple linear equations. They can be quadratic, exponential, trigonometric, and more. Each type of function has its unique properties and applications. By understanding the basic concepts of functions, you'll be well-equipped to tackle more complex mathematical problems and real-world scenarios. So, remember, a function is a rule that takes an input, applies some operations, and produces an output. This concept is fundamental to mathematics and is used extensively in various fields, from physics to computer science.

In our case, the rule is: "Take the input x, multiply it by 2/3, and then subtract 12." So, if we put in a 36, the function will do exactly that! Now, let's move on to the next step: actually plugging in the number and doing the math.

Plugging in x = 36

Okay, now for the fun part! We need to evaluate f(36). What this means is that wherever we see an x in our function, we're going to replace it with 36. It's like substituting a player on a sports team – we're swapping x out for 36. Mastering substitution is a critical step in evaluating functions. It's like the first step in following a recipe – you need to gather the right ingredients before you can start cooking. The concept of substitution extends beyond simple function evaluation; it's used in solving equations, simplifying expressions, and even in more advanced mathematical topics like calculus. Think of it as a fundamental skill that unlocks a wide range of mathematical abilities. For a solid understanding, practice substituting different values into various functions. Try plugging in negative numbers, fractions, and even algebraic expressions. The more you practice, the more comfortable you'll become with this essential technique. Also, pay attention to the order of operations (PEMDAS/BODMAS) to ensure you're performing the calculations correctly. Remember, parentheses/brackets first, then exponents/orders, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). Keeping these rules in mind will help you avoid common errors and achieve accurate results. So, substitution is not just about replacing a variable with a value; it's about understanding how to manipulate expressions and maintain mathematical integrity. It's a foundational skill that will serve you well in your mathematical journey.

So, let's rewrite our function f(x) = (2/3)x - 12, replacing x with 36:

f(36) = (2/3) * 36 - 12

See? We just swapped the x for 36. Now we have a simple arithmetic problem to solve. Let's move on to the next step: solving this equation.

Solving the Equation

Alright, we've plugged in 36, and now we have f(36) = (2/3) * 36 - 12. To solve this, we need to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This principle is not just a mathematical rule; it's a fundamental framework for approaching problem-solving in various contexts. Understanding the order of operations ensures consistency and accuracy in calculations, whether you're balancing your budget or designing a complex engineering structure. Think of it as a universal language for mathematical operations, ensuring everyone interprets the expression the same way. To deepen your understanding, consider exploring the history and evolution of mathematical notation and conventions. Understanding why we use certain symbols and rules can provide a richer appreciation for the elegance and logic of mathematics. Also, remember that the order of operations is not arbitrary; it's based on mathematical principles that ensure consistency and avoid ambiguity. For instance, multiplication and division have higher precedence than addition and subtraction because they represent repeated addition and subtraction, respectively. This hierarchy reflects the underlying mathematical structure and allows us to simplify complex expressions in a systematic way. So, when tackling any mathematical problem, always keep the order of operations in mind. It's the key to unlocking accurate and meaningful solutions.

First up is the multiplication: (2/3) * 36. We can think of this as (2 * 36) / 3. So, 2 * 36 is 72, and then we divide that by 3.

72 / 3 = 24

So, now we have:

f(36) = 24 - 12

Much simpler, right? Now, we just need to subtract 12 from 24.

24 - 12 = 12

So, f(36) = 12! We did it! Let's recap our steps in the next section.

Recap and Final Answer

Awesome work, guys! We've successfully evaluated f(36) for the function f(x) = (2/3)x - 12. Let's quickly recap the steps we took to get there. Regular recaps are crucial for reinforcing learning and ensuring a thorough understanding of the material. Think of it as building a mental map of the problem-solving process. By revisiting the steps, you solidify the connections between concepts and develop a stronger grasp of the overall approach. Recaps are not just about memorizing procedures; they're about understanding the logic and reasoning behind each step. They help you identify patterns, recognize potential pitfalls, and develop strategies for tackling similar problems in the future. To make your recaps even more effective, try summarizing the key steps in your own words. This active recall process strengthens memory and helps you internalize the concepts. Also, consider drawing diagrams or creating flowcharts to visually represent the problem-solving process. Visual aids can enhance understanding and make it easier to recall the steps later. Furthermore, recaps provide an opportunity to reflect on the problem-solving experience and identify areas for improvement. Ask yourself questions like: What were the most challenging steps? What strategies did I find most helpful? How could I approach this problem more efficiently in the future? By engaging in this reflective practice, you'll develop a deeper understanding of your own learning style and become a more effective problem-solver. So, remember, recaps are not just an end-of-lesson activity; they're an integral part of the learning process that helps you build a solid foundation for future success.

1. Understood the function: We knew that f(x) = (2/3)x - 12 meant we multiply the input by 2/3 and then subtract 12.
2. Plugged in x = 36: We substituted 36 for x in the function: f(36) = (2/3) * 36 - 12.
3. Solved the equation: We followed the order of operations, first multiplying (2/3) * 36 to get 24, and then subtracting 12.

So, our final answer is:

f(36) = 12

Great job, everyone! You've successfully evaluated a function. Keep practicing, and you'll become a function-evaluating pro in no time!

Practice Problems

Want to test your skills further? Here are a couple of practice problems you can try:

1. Evaluate g(x) = 5x + 7 for x = -2.
2. Evaluate h(x) = x^2 - 3x + 1 for x = 4.

Working through these practice problems will help you solidify your understanding of function evaluation and build confidence in your abilities. Remember, the key to mastering any mathematical skill is practice, practice, practice! Don't be afraid to make mistakes – they're a valuable part of the learning process. Each error provides an opportunity to identify areas for improvement and refine your problem-solving strategies. To make your practice sessions even more effective, try varying the types of problems you tackle. Explore different types of functions, such as linear, quadratic, and exponential functions. Also, try problems with varying levels of difficulty, from simple substitutions to more complex manipulations. This will help you develop a versatile skill set and prepare you for a wide range of mathematical challenges. Furthermore, consider working through practice problems with a friend or study group. Explaining your thought process to others can help you clarify your understanding and identify any gaps in your knowledge. Also, learning from others' approaches and perspectives can broaden your problem-solving toolkit. So, grab a pen and paper, dive into these practice problems, and keep building your function evaluation skills. With consistent effort and a positive attitude, you'll be well on your way to becoming a math whiz!

Good luck, and happy calculating!