Evaluate F(j, K) = 3j + 2k For J=5 And K=9 A Step-by-Step Solution

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Introduction

In the realm of mathematics, functions play a pivotal role in describing relationships between variables. Evaluating functions is a fundamental skill that allows us to determine the output of a function for specific input values. This article will delve into the process of evaluating a function with two variables, using the example of $f(j, k) = 3j + 2k$. We will explore how to substitute given values for the variables and simplify the expression to arrive at the solution. This comprehensive guide will walk you through each step, ensuring a clear understanding of the process. Understanding function evaluation is crucial for various mathematical applications, from simple algebra to more complex calculus problems. By mastering this skill, you'll be better equipped to tackle a wide range of mathematical challenges and real-world applications. This article aims to provide a detailed explanation, making it easy for anyone to grasp the concept and apply it confidently. Whether you're a student learning algebra or someone looking to refresh your math skills, this guide will offer a solid foundation in function evaluation.

Functions are the building blocks of mathematical models, representing how one quantity depends on others. In many situations, these relationships involve more than one variable. For instance, the area of a rectangle depends on both its length and width, or the temperature felt by a person can depend on both the actual air temperature and the wind speed. Understanding how to work with functions of multiple variables is crucial in various fields, including physics, engineering, economics, and computer science. Functions are not just abstract mathematical concepts; they are tools that help us describe and predict real-world phenomena. For example, in economics, a production function might describe how the output of a company depends on the amount of labor and capital invested. In physics, the force of gravity between two objects depends on their masses and the distance between them. The ability to evaluate functions with multiple variables allows us to make quantitative predictions and understand how changes in one variable can affect the others. This article will focus specifically on the mechanics of evaluating a function with two variables, providing a clear and methodical approach that can be applied to a variety of problems.

Understanding the Function Notation

Before we dive into the evaluation process, let's clarify the function notation used in the problem. The expression $f(j, k) = 3j + 2k$ represents a function named "f" that takes two variables, "j" and "k", as inputs. The output of the function is determined by the expression $3j + 2k$. This means that to find the value of the function for specific values of j and k, we need to substitute those values into the expression. Function notation is a shorthand way of expressing a mathematical relationship. The function name, in this case "f", is followed by the input variables in parentheses. The expression on the right side of the equals sign defines how the function operates on the inputs. Understanding this notation is essential for interpreting and working with functions effectively. It's a language that mathematicians and scientists use to communicate complex ideas concisely. The use of function notation also allows us to easily distinguish between different functions and their respective variables. For example, we might have another function, g(x, y), which represents a different relationship between the variables x and y. By using different function names, we can avoid confusion and keep our mathematical expressions clear and organized. In the given example, the function f(j, k) takes two inputs, but functions can also have one input, three inputs, or even more. The key is that the function notation provides a structured way to represent the relationship between inputs and outputs.

Step-by-Step Evaluation: Substituting the Values

The problem asks us to evaluate the function when $j = 5$ and $k = 9$. This means we need to substitute these values into the expression $3j + 2k$. Let's break down the substitution process step by step:

  1. Replace 'j' with 5: The term $3j$ becomes $3 * 5$.
  2. Replace 'k' with 9: The term $2k$ becomes $2 * 9$.
  3. The expression now looks like this: $f(5, 9) = 3 * 5 + 2 * 9$.

This substitution is the core of evaluating the function. We are essentially replacing the symbolic variables with their numerical values. It's important to be meticulous in this step to avoid errors. Make sure you are substituting the correct values for the corresponding variables. Double-checking your substitutions can save you from making mistakes later on. The process of substitution is not unique to this particular function; it's a fundamental technique that applies to all functions, regardless of their complexity. Whether the function involves simple arithmetic operations or more advanced mathematical concepts, the basic principle of substitution remains the same. By mastering this step, you'll be well-prepared to tackle a wide range of function evaluation problems. The substitution step transforms the symbolic expression into a numerical one, making it ready for the next stage: simplification.

Step-by-Step Evaluation: Simplifying the Expression

Now that we have substituted the values, we need to simplify the expression $3 * 5 + 2 * 9$. Remember 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). In this case, we have multiplication and addition. Multiplication comes before addition, so we perform the multiplications first:

  1. 3∗5=153 * 5 = 15

  2. 2∗9=182 * 9 = 18

  3. The expression now becomes: $f(5, 9) = 15 + 18$
  4. Finally, we perform the addition: $15 + 18 = 33$

Therefore, $f(5, 9) = 33$.

Simplifying the expression involves applying the correct order of operations to arrive at a single numerical value. This step is just as crucial as the substitution step. Making an error in the order of operations can lead to an incorrect result. It's a good practice to write out each step clearly, as we have done here, to minimize the chances of mistakes. The order of operations is a fundamental principle in mathematics, ensuring that everyone arrives at the same answer when evaluating an expression. Without a consistent order, the same expression could be interpreted in multiple ways, leading to confusion and inconsistency. By following the PEMDAS/BODMAS rule, we can maintain clarity and accuracy in our calculations. The simplification step is not just about arithmetic; it's about applying mathematical conventions to ensure that our expressions are evaluated correctly. In more complex functions, the simplification process might involve multiple steps, including combining like terms, factoring, or using other algebraic techniques. However, the basic principle of following the order of operations remains the same. The final result, 33, represents the value of the function f(j, k) when j is 5 and k is 9. This single number encapsulates the output of the function for the given inputs.

Final Answer and Conclusion

We have successfully evaluated the function $f(j, k) = 3j + 2k$ when $j = 5$ and $k = 9$. By substituting the given values and simplifying the expression, we found that $f(5, 9) = 33$. This means that the output of the function f when the inputs are 5 and 9 is 33.

In conclusion, evaluating functions with two variables involves a straightforward process: substitute the given values for the variables and simplify the resulting expression. This example demonstrates the fundamental principles of function evaluation, which can be applied to a wide variety of functions and mathematical problems. Mastering this skill is essential for anyone studying mathematics or working in fields that rely on mathematical modeling. The ability to evaluate functions accurately and efficiently is a key step in solving more complex problems and understanding the relationships between variables. This article has provided a detailed and step-by-step guide to the process, ensuring that you have a solid understanding of the concepts involved. From understanding the function notation to substituting values and simplifying expressions, each step is crucial in arriving at the correct answer. By practicing these steps, you can build confidence in your ability to evaluate functions and tackle more challenging mathematical tasks. The importance of function evaluation extends beyond the classroom; it's a practical skill that is used in many real-world applications. Whether you are calculating the trajectory of a projectile, modeling population growth, or analyzing financial data, the ability to work with functions is essential for making informed decisions and solving problems effectively. The result we obtained, 33, is not just a number; it's a piece of information that represents the output of the function under specific conditions. This output can be used for further analysis, comparison, or prediction, depending on the context of the problem.

  • Function Evaluation: Evaluating functions involves substituting given values for variables and simplifying the expression.
  • Function Notation: Understand the notation f(j, k) = 3j + 2k, where f is the function name, j and k are input variables, and 3j + 2k is the expression that defines the function.
  • Substitution: Replace variables with their numerical values (e.g., j with 5 and k with 9).
  • Order of Operations: Follow the order of operations (PEMDAS/BODMAS) to simplify the expression correctly.
  • Simplification: Perform multiplication before addition: 3 * 5 = 15 and 2 * 9 = 18.
  • Final Calculation: Add the results: 15 + 18 = 33.
  • Result: The function value f(5, 9) is 33.