Evaluating G(x) = 3x^2 - 1 When X = -2 A Step-by-Step Guide
In the realm of mathematics, evaluating functions is a fundamental skill. A function, in its essence, is a mathematical rule that assigns a unique output value for each input value. This concept is crucial in various areas of mathematics and its applications. When we talk about evaluating a function, we mean finding the output value of the function for a specific input value. This process involves substituting the given input value into the function's expression and then simplifying the expression to obtain the output. In this article, we will delve into the process of evaluating the function g(x) = 3x^2 - 1 when x = -2. This example will provide a clear and concise illustration of how to evaluate a function for a given input. Understanding this process is essential for mastering more complex mathematical concepts and problem-solving techniques. The ability to evaluate functions accurately and efficiently is a cornerstone of mathematical proficiency. So, let's embark on this journey of understanding function evaluation, which will undoubtedly enhance your mathematical skills and broaden your understanding of the mathematical world.
Understanding the Function Notation
Before we dive into the evaluation, let's first understand the notation we're dealing with. The expression g(x) = 3x^2 - 1 represents a function named 'g'. The 'x' inside the parentheses is the input variable, and the expression '3x^2 - 1' defines the rule that the function applies to the input. This rule tells us how to transform the input 'x' into the output value. The left-hand side, g(x), represents the output of the function when the input is 'x'. This notation is a concise and powerful way to represent functions and their behavior. The function notation allows us to clearly express the relationship between the input and the output, making it easier to analyze and manipulate mathematical expressions. Understanding this notation is crucial for working with functions in mathematics, as it forms the basis for many concepts and calculations. The function g(x) is a quadratic function because the highest power of x is 2. Quadratic functions are known for their parabolic shapes when graphed, and they have numerous applications in physics, engineering, and other fields. By understanding the function notation, we can effectively communicate and work with mathematical ideas, paving the way for deeper exploration of mathematical concepts and problem-solving. So, let's keep this notation in mind as we proceed to evaluate the function for a specific input value.
Substituting the Value of x
Now that we understand the function notation, the next step is to substitute the given value of x, which is -2, into the function's expression. This means we replace every instance of 'x' in the expression '3x^2 - 1' with '-2'. This substitution is the heart of the evaluation process, as it allows us to apply the function's rule to the specific input value. After the substitution, we obtain the expression '3(-2)^2 - 1'. It's crucial to pay attention to the order of operations when substituting values, especially when dealing with exponents and negative numbers. Substituting the value correctly ensures that we are following the function's rule accurately and will arrive at the correct output. This step requires careful attention to detail, as a small mistake in the substitution can lead to an incorrect result. The substitution process is not only essential for evaluating functions but also for solving equations and inequalities. By substituting values, we can determine whether a particular value satisfies an equation or inequality. This technique is widely used in various mathematical contexts and is a fundamental skill for any mathematics student. So, let's proceed with the substitution and make sure we replace every 'x' with '-2' to accurately reflect the function's rule for the given input.
Following the Order of Operations (PEMDAS/BODMAS)
After substituting x = -2 into the function, we have the expression 3(-2)^2 - 1. To simplify this expression, we must adhere to the order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This order ensures that we perform the operations in the correct sequence, leading to the accurate evaluation of the expression. First, we address the exponent: (-2)^2 means (-2) multiplied by itself, which equals 4. So, the expression becomes 3 * 4 - 1. Next, we perform the multiplication: 3 * 4 equals 12. Now, the expression is simplified to 12 - 1. Finally, we perform the subtraction: 12 - 1 equals 11. By meticulously following the order of operations, we have successfully simplified the expression and found the output value of the function. The order of operations is a cornerstone of mathematical calculations, ensuring consistency and accuracy in our results. It's essential to master this concept to avoid errors in evaluating expressions and solving equations. So, always remember PEMDAS/BODMAS and apply it diligently when simplifying mathematical expressions.
Simplifying the Expression
Following the order of operations, we now simplify the expression 3(-2)^2 - 1 step by step. First, we evaluate the exponent: (-2)^2 = 4. This means we multiply -2 by itself, resulting in a positive 4. The expression now becomes 3 * 4 - 1. Next, we perform the multiplication: 3 * 4 = 12. The expression is further simplified to 12 - 1. Finally, we perform the subtraction: 12 - 1 = 11. This final result is the output value of the function g(x) when x = -2. Simplifying expressions is a crucial skill in mathematics, as it allows us to break down complex expressions into simpler, more manageable forms. By following the order of operations and applying basic arithmetic operations, we can accurately simplify expressions and solve mathematical problems. This process not only helps us find the correct answer but also enhances our understanding of the underlying mathematical concepts. The ability to simplify expressions is essential for various areas of mathematics, including algebra, calculus, and trigonometry. It is a fundamental skill that students need to develop to succeed in their mathematical studies. So, let's practice simplifying expressions regularly to strengthen our mathematical abilities and problem-solving skills.
The Final Result
After meticulously substituting x = -2 into the function g(x) = 3x^2 - 1 and simplifying the expression following the order of operations, we arrive at the final result: g(-2) = 11. This means that when the input value is -2, the output value of the function is 11. This result represents a specific point on the graph of the function g(x). The point is (-2, 11), where -2 is the x-coordinate (input) and 11 is the y-coordinate (output). This ordered pair is a solution to the function, indicating the relationship between the input and output values. The final result, 11, is the answer to our evaluation problem. It demonstrates the function's behavior at a particular point and provides valuable information about the function's overall characteristics. Understanding how to obtain and interpret these results is essential for analyzing and applying functions in various mathematical and real-world contexts. Function evaluation is a fundamental skill that underlies many mathematical concepts and applications. By mastering this skill, we can confidently tackle more complex problems and gain a deeper understanding of the mathematical world. So, let's embrace the power of function evaluation and continue to explore the fascinating world of mathematics.
In conclusion, we have successfully evaluated the function g(x) = 3x^2 - 1 when x = -2, arriving at the result g(-2) = 11. This process involved substituting the given value of x into the function's expression, simplifying the expression following the order of operations, and obtaining the final output value. Function evaluation is a cornerstone of mathematics, serving as a fundamental skill for understanding and applying mathematical concepts. It allows us to determine the output of a function for a specific input, providing valuable insights into the function's behavior and characteristics. The ability to evaluate functions accurately and efficiently is crucial for solving equations, graphing functions, and analyzing mathematical models. Function evaluation is not only essential in mathematics but also in various fields such as physics, engineering, computer science, and economics, where mathematical models are used to represent real-world phenomena. By mastering function evaluation, we equip ourselves with a powerful tool for problem-solving and critical thinking. This skill enables us to analyze and interpret mathematical relationships, making informed decisions and predictions. So, let's continue to practice and refine our function evaluation skills, as they will undoubtedly serve us well in our mathematical journey and beyond.
