Evaluating Functions F(x) And G(x) At Specific Points

In mathematics, functions are essential tools for describing relationships between variables. This article delves into the evaluation of two specific functions, $f(x) = 3x^2 - 3x$ and $g(x) = -5x + 4$, at particular points. Understanding how to evaluate functions is a fundamental skill in algebra and calculus, paving the way for more advanced mathematical concepts. We will walk through the step-by-step process of substituting values into these functions and simplifying the results. This process will enhance your understanding of function notation and algebraic manipulation. This article will provide you with a clear understanding of how to evaluate functions. By the end of this guide, you'll be equipped with the knowledge and skills to tackle similar problems with confidence. Function evaluation is a key concept that appears in various fields, from physics and engineering to economics and computer science. Mastering this skill will not only help you in your mathematical studies but also in real-world applications. Let's dive into the specifics of evaluating $f(x)$ and $g(x)$ at given points and explore the significance of these evaluations.

Evaluating f(x) at x = -2

Let's begin by evaluating the function f(x) = 3x² - 3x at the point x = -2. To do this, we substitute -2 for x in the function's expression. This substitution process is a fundamental step in evaluating any function at a specific point. It involves replacing the variable x with the given numerical value and then simplifying the resulting expression. This is where the core of function evaluation lies. By performing this substitution carefully, we can determine the output of the function for the given input. The function f(x) essentially defines a rule for transforming input values (x) into output values (f(x)). By substituting -2 for x, we're applying this rule to a specific input and finding its corresponding output. This process helps us understand how the function behaves and what values it produces for different inputs. It's a crucial skill for analyzing and interpreting mathematical models in various contexts. Here’s the step-by-step calculation:

1. Substitute x with -2: Replace every instance of x in the function’s formula with -2. This gives us f(-2) = 3(-2)² - 3(-2). The parentheses are crucial here as they ensure that the negative sign is handled correctly during the squaring operation and multiplication. Incorrect handling of the negative sign can lead to errors in the final result. This is a common pitfall for students learning function evaluation, so paying close attention to this step is vital.
2. Simplify the expression: Now, we simplify the expression following the order of operations (PEMDAS/BODMAS). First, we calculate the exponent: (-2)² = 4. This squaring operation is a crucial part of the calculation and must be done before multiplication or subtraction. Next, we perform the multiplications: 3 * 4 = 12 and -3 * -2 = 6. Remember that multiplying two negative numbers results in a positive number. Finally, we perform the addition: 12 + 6 = 18. Thus, we find that f(-2) = 18.

Therefore, the value of the function f(x) at x = -2 is 18. This means that when we input -2 into the function, the function outputs 18. This result gives us a specific point on the graph of the function, which can be useful for visualizing and analyzing the function's behavior. Understanding how to correctly substitute and simplify expressions is crucial for accurate function evaluation. This process allows us to determine the function's output for a given input, providing valuable insights into the function's behavior and properties.

Evaluating g(x) at x = 6

Next, let's evaluate the function g(x) = -5x + 4 at the point x = 6. Similar to the previous example, we substitute 6 for x in the function's expression. This substitution is a key step in determining the function's output for the given input. The function g(x) defines a linear relationship between x and g(x), and by substituting x = 6, we can find the corresponding value of g(6). This process is essential for understanding the function's behavior and its values at specific points. Function evaluation is a fundamental concept in mathematics, and mastering this skill is crucial for solving various problems in algebra and calculus. By carefully substituting the value and simplifying the expression, we can accurately determine the function's output.

1. Substitute x with 6: Replace x in the function’s formula with 6. This gives us g(6) = -5(6) + 4. Again, using parentheses is important to ensure the correct order of operations and to avoid sign errors. Substituting the value correctly sets the stage for accurate simplification and ultimately, the correct function evaluation. This is a critical step in the process, as any error here will propagate through the rest of the calculation.
2. Simplify the expression: Following the order of operations, we first perform the multiplication: -5 * 6 = -30. Then, we perform the addition: -30 + 4 = -26. Therefore, we find that g(6) = -26. This result means that when we input 6 into the function g(x), the function outputs -26. This tells us a specific point on the line represented by the function g(x). Understanding how to simplify expressions correctly is crucial for accurate function evaluation. This process involves applying the rules of arithmetic and algebra to arrive at the final result.

Thus, the value of the function g(x) at x = 6 is -26. This means that the point (6, -26) lies on the graph of the function g(x). The result of -26 indicates the output or y-value of the function when the input or x-value is 6. This point is part of the linear relationship defined by the function g(x) = -5x + 4. Understanding how to evaluate functions at specific points is fundamental for graphing functions, solving equations, and analyzing mathematical models.

In conclusion, we have successfully evaluated the functions f(x) = 3x² - 3x at x = -2 and g(x) = -5x + 4 at x = 6. We found that f(-2) = 18 and g(6) = -26. These evaluations demonstrate the fundamental process of substituting values into functions and simplifying the resulting expressions. The ability to accurately evaluate functions is a crucial skill in mathematics, forming the basis for more advanced concepts such as calculus and analysis. This article has provided a step-by-step guide to function evaluation, highlighting the importance of correct substitution and simplification. Mastering this skill will significantly enhance your mathematical proficiency and problem-solving abilities.

Understanding function evaluation is not just an academic exercise; it has practical applications in various fields. For example, in physics, functions are used to model motion, and evaluating these functions at specific times allows us to determine the position or velocity of an object. In economics, functions can represent cost or revenue, and evaluating them can help businesses make informed decisions. In computer science, functions are the building blocks of programs, and understanding how they work is essential for software development. The process of function evaluation involves substituting a given value for the variable in the function's expression and then simplifying the expression using the order of operations. This requires careful attention to detail and a solid understanding of algebraic principles. By practicing function evaluation, you can improve your mathematical skills and gain confidence in your ability to solve complex problems.

By mastering the techniques discussed in this article, you can confidently approach function evaluation problems and apply these skills in various mathematical and real-world contexts. Remember, practice is key to improving your skills, so try evaluating different functions at various points to solidify your understanding. The more you practice, the more comfortable you will become with the process, and the better you will be able to apply these skills in more complex situations. Function evaluation is a foundational concept in mathematics, and a strong understanding of this topic will serve you well in your future studies and career.