Evaluating The Function F(x) = 3x² - 2x For Domain 1 2 3
Introduction to Function Evaluation
In the realm of mathematics, function evaluation is a fundamental process. It involves substituting specific values from the domain of a function into the function's expression and then calculating the resulting output. This process allows us to understand the behavior of the function across different inputs and is crucial in various mathematical applications, including graphing, solving equations, and modeling real-world phenomena. Understanding function evaluation is essential for anyone delving into algebra, calculus, and beyond. A function, in its essence, is a mapping between a set of inputs (the domain) and a set of possible outputs (the range). The function is usually denoted by a letter, such as f, g, or h, and the input variable is often represented by x. The expression that defines the function specifies how the input is transformed into the output. For example, the function f(x) = 3x² - 2x defines a specific relationship between the input x and the output, which is calculated by squaring x, multiplying by 3, and then subtracting 2 times x. To evaluate a function, we replace the input variable x with a specific value from the domain. This substitution creates an expression that can be simplified to yield the corresponding output. For instance, if we want to evaluate f(x) = 3x² - 2x at x = 1, we substitute 1 for x in the expression, resulting in f(1) = 3(1)² - 2(1). This expression can then be simplified to find the value of the function at x = 1. The domain of a function is the set of all possible input values for which the function is defined. In simpler terms, it's the collection of x-values that you can plug into the function without causing any mathematical errors, such as division by zero or taking the square root of a negative number. The domain can be explicitly specified, as in the case of the set {1, 2, 3} for the function we are evaluating, or it can be implied by the function's expression. Understanding the domain is crucial because it tells us the range of inputs for which the function produces valid outputs. Different types of functions have different domain considerations. For example, polynomial functions, like the one we're evaluating, generally have a domain of all real numbers, meaning any real number can be used as an input. However, rational functions (functions with a fraction where the denominator contains a variable) have restrictions on their domain because the denominator cannot be zero. Similarly, functions involving square roots have the restriction that the expression under the square root must be non-negative. The process of function evaluation often involves several steps. First, you identify the function and the input value you want to evaluate. Second, you substitute the input value for the variable in the function's expression. Third, you simplify the expression using the order of operations (PEMDAS/BODMAS), which dictates the sequence in which mathematical operations are performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Fourth, the result of the simplification is the output value of the function for the given input. Function evaluation is not just a theoretical exercise; it has numerous practical applications. In graphing, we evaluate functions at several points in their domain to plot the corresponding points on a coordinate plane. Connecting these points gives us a visual representation of the function's behavior. In solving equations, we often evaluate functions to check potential solutions. For instance, if we have an equation f(x) = 0, we can evaluate f(x) at a potential solution to see if it indeed results in 0. In modeling real-world phenomena, functions are used to represent relationships between different variables. Evaluating these functions allows us to make predictions and understand how the system being modeled behaves. For example, a function might model the population growth of a species over time. By evaluating the function at different time points, we can estimate the population size at those times. In conclusion, function evaluation is a cornerstone of mathematical understanding. It is the process of substituting values from the domain into a function's expression and calculating the corresponding output. This process is essential for graphing, solving equations, modeling, and numerous other applications. Understanding the domain of a function is crucial for determining the valid inputs, and the order of operations must be carefully followed when simplifying the expression. With a solid grasp of function evaluation, you can unlock the power of functions to represent and analyze mathematical relationships in a wide range of contexts.
Evaluating f(x) = 3x² - 2x for x = 1
To begin our evaluation, let's consider the first value in the domain, x = 1. We need to substitute this value into the function f(x) = 3x² - 2x and simplify the expression. This process involves replacing every instance of x in the function's expression with the value 1. This step is crucial in function evaluation as it sets the stage for the subsequent calculations that will lead to the output value of the function for the given input. Proper substitution ensures that the rest of the evaluation is accurate. Mistakes in substitution can lead to incorrect results, so care and precision are essential at this stage. When we substitute x = 1 into the function f(x) = 3x² - 2x, we get f(1) = 3(1)² - 2(1). This new expression now contains only numbers and mathematical operations. The next step is to simplify this expression using the order of operations, often remembered by the acronym 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 correct final result. In the expression 3(1)² - 2(1), the first operation we encounter according to PEMDAS/BODMAS is the exponent. We have (1)², which means 1 raised to the power of 2. This is simply 1 multiplied by itself, which equals 1. So, (1)² = 1. Substituting this back into our expression, we now have 3(1) - 2(1). The next operations in the order are multiplication and division, which we perform from left to right. In our expression, we have two multiplication operations: 3(1) and 2(1). Multiplying 3 by 1 gives us 3, and multiplying 2 by 1 gives us 2. So, the expression becomes 3 - 2. Finally, we perform the addition and subtraction operations, again from left to right. In this case, we have only one operation left: subtraction. Subtracting 2 from 3 gives us 1. Therefore, 3 - 2 = 1. This is the final result of the simplification. Putting it all together, we have f(1) = 3(1)² - 2(1) = 3(1) - 2 = 3 - 2 = 1. This means that when the input to the function f(x) = 3x² - 2x is 1, the output is 1. We have successfully evaluated the function at x = 1. This single evaluation provides one data point for the function, which can be used for graphing or other analytical purposes. The process of function evaluation may seem straightforward, but it is a fundamental skill in mathematics and is used extensively in more advanced topics. Understanding how to correctly substitute and simplify expressions is crucial for success in algebra, calculus, and beyond. The careful application of the order of operations is paramount to avoid errors. In summary, evaluating f(x) = 3x² - 2x at x = 1 involves substituting 1 for x, simplifying the resulting expression using the order of operations, and arriving at the output value of 1. This process demonstrates the basic mechanics of function evaluation and lays the groundwork for evaluating the function at other points in its domain.
Evaluating f(x) = 3x² - 2x for x = 2
Now, let's proceed to the second value in our domain, x = 2. As we did with x = 1, we will substitute this value into the function f(x) = 3x² - 2x and simplify the resulting expression. This step is another iteration of the function evaluation process, reinforcing the method and illustrating how the output of the function changes with different inputs. By evaluating the function at multiple points, we can begin to understand its overall behavior and characteristics. Substituting x = 2 into the function f(x) = 3x² - 2x, we replace every instance of x with 2, resulting in f(2) = 3(2)² - 2(2). This new expression is now ready for simplification. It contains numerical values and mathematical operations that, when performed in the correct order, will yield the output value of the function at x = 2. The accuracy of the substitution is paramount, as any error at this stage will propagate through the rest of the evaluation, leading to an incorrect result. Therefore, double-checking the substitution is a good practice to ensure precision. Once we have the expression 3(2)² - 2(2), we apply the order of operations (PEMDAS/BODMAS) to simplify it. The first operation we encounter is the exponent. We have (2)², which means 2 raised to the power of 2. This is 2 multiplied by itself, which equals 4. So, (2)² = 4. Substituting this back into our expression, we now have 3(4) - 2(2). This step reduces the complexity of the expression and moves us closer to the final result. By correctly evaluating the exponent, we maintain the integrity of the mathematical process and ensure the accuracy of the outcome. The next operations in the order are multiplication and division, which we perform from left to right. In our expression 3(4) - 2(2), we have two multiplication operations: 3(4) and 2(2). Multiplying 3 by 4 gives us 12, and multiplying 2 by 2 gives us 4. So, the expression becomes 12 - 4. Performing the multiplications simplifies the expression further and sets the stage for the final subtraction. Each multiplication is a straightforward arithmetic operation, but they are crucial in determining the correct output of the function at x = 2. The final operation to perform is subtraction. We have 12 - 4, which means subtracting 4 from 12. This results in 8. Therefore, 12 - 4 = 8. This is the final result of the simplification. Putting it all together, we have f(2) = 3(2)² - 2(2) = 3(4) - 4 = 12 - 4 = 8. This means that when the input to the function f(x) = 3x² - 2x is 2, the output is 8. We have successfully evaluated the function at x = 2. This gives us another data point for the function, which, along with the point we found for x = 1, helps us to visualize the function's behavior. The process of function evaluation is consistent across different input values. The key steps are substitution and simplification using the order of operations. By carefully following these steps, we can accurately determine the output of the function for any given input. In summary, evaluating f(x) = 3x² - 2x at x = 2 involves substituting 2 for x, simplifying the resulting expression using the order of operations, and arriving at the output value of 8. This process reinforces the method of function evaluation and demonstrates how the function's output changes as the input changes.
Evaluating f(x) = 3x² - 2x for x = 3
Finally, let's evaluate the function for the last value in our domain, x = 3. Following the same procedure as before, we will substitute x = 3 into the function f(x) = 3x² - 2x and simplify the expression. This third evaluation will give us a more comprehensive understanding of the function's behavior over the specified domain. By this point, the process of function evaluation should be becoming more familiar, and the consistent application of the steps will lead to an accurate result. Substituting x = 3 into the function f(x) = 3x² - 2x, we replace every instance of x with 3, which gives us f(3) = 3(3)² - 2(3). This expression is now set up for simplification. It consists of numerical values and mathematical operations that need to be performed in the correct order to obtain the function's output at x = 3. The substitution step is critical, and its accuracy is paramount for the validity of the subsequent calculations. Any error in this step will lead to an incorrect final result. Therefore, a careful check of the substitution is always recommended. With the expression 3(3)² - 2(3) in hand, we proceed to simplify it using the order of operations (PEMDAS/BODMAS). The first operation we encounter is the exponent. We have (3)², which means 3 raised to the power of 2. This is 3 multiplied by itself, which equals 9. So, (3)² = 9. Substituting this back into our expression, we now have 3(9) - 2(3). Evaluating the exponent is a key step in simplifying the expression, and doing it correctly ensures that the subsequent operations are performed on the correct values. This step reduces the complexity of the expression and brings us closer to the final result. Next in the order of operations are multiplication and division, which we perform from left to right. In our expression 3(9) - 2(3), we have two multiplication operations: 3(9) and 2(3). Multiplying 3 by 9 gives us 27, and multiplying 2 by 3 gives us 6. So, the expression becomes 27 - 6. Performing these multiplications simplifies the expression further, setting the stage for the final subtraction. Each multiplication is a basic arithmetic operation, but they are essential for determining the correct output of the function at x = 3. The final operation is subtraction. We have 27 - 6, which means subtracting 6 from 27. This results in 21. Therefore, 27 - 6 = 21. This is the final result of the simplification. Putting it all together, we have f(3) = 3(3)² - 2(3) = 3(9) - 6 = 27 - 6 = 21. This means that when the input to the function f(x) = 3x² - 2x is 3, the output is 21. We have successfully evaluated the function at x = 3. Now we have three data points for the function, corresponding to the inputs 1, 2, and 3. These points provide a good sense of the function's behavior over the specified domain and can be used for graphing or other analytical purposes. The process of function evaluation has been consistently applied across the three input values, demonstrating the method's reliability and accuracy. The key steps remain the same: substitution of the input value into the function's expression, followed by simplification using the order of operations. By following these steps carefully, we can confidently determine the output of the function for any given input. In summary, evaluating f(x) = 3x² - 2x at x = 3 involves substituting 3 for x, simplifying the resulting expression using the order of operations, and arriving at the output value of 21. This final evaluation completes our analysis of the function over the domain {1, 2, 3} and provides a clear picture of its behavior within this range.
Summary of Function Evaluation for the Domain {1, 2, 3}
Having evaluated the function f(x) = 3x² - 2x for each value in the domain {1, 2, 3}, we can now summarize our findings. This summary will provide a concise overview of the function's output for each input value and will help to illustrate the function's behavior over the given domain. The process of function evaluation involves substituting each value from the domain into the function's expression and simplifying the resulting expression using the order of operations. This process allows us to determine the output value of the function for each input value, providing a set of ordered pairs that can be used for graphing, analysis, or other applications. For x = 1, we found that f(1) = 1. This means that when the input to the function is 1, the output is also 1. This gives us the ordered pair (1, 1), which represents a point on the graph of the function. The process of function evaluation at x = 1 involved substituting 1 for x in the expression 3x² - 2x, which resulted in 3(1)² - 2(1). Simplifying this expression using the order of operations, we first evaluated the exponent, (1)² = 1, then performed the multiplications, 3(1) = 3 and 2(1) = 2, and finally performed the subtraction, 3 - 2 = 1. This process demonstrates the step-by-step application of the order of operations in function evaluation. For x = 2, we found that f(2) = 8. This means that when the input to the function is 2, the output is 8. This gives us the ordered pair (2, 8), another point on the graph of the function. Evaluating the function at x = 2 involved substituting 2 for x in the expression 3x² - 2x, resulting in 3(2)² - 2(2). Simplifying this expression, we first evaluated the exponent, (2)² = 4, then performed the multiplications, 3(4) = 12 and 2(2) = 4, and finally performed the subtraction, 12 - 4 = 8. This evaluation further illustrates the methodical process of function evaluation and the importance of the order of operations. For x = 3, we found that f(3) = 21. This means that when the input to the function is 3, the output is 21. This gives us the ordered pair (3, 21), yet another point on the graph of the function. The evaluation of the function at x = 3 involved substituting 3 for x in the expression 3x² - 2x, which resulted in 3(3)² - 2(3). Simplifying this expression, we first evaluated the exponent, (3)² = 9, then performed the multiplications, 3(9) = 27 and 2(3) = 6, and finally performed the subtraction, 27 - 6 = 21. This final evaluation provides a clear picture of the function's behavior over the specified domain and reinforces the importance of accurate substitution and simplification. In summary, for the domain {1, 2, 3}, the function f(x) = 3x² - 2x yields the following outputs: f(1) = 1, f(2) = 8, and f(3) = 21. These three ordered pairs, (1, 1), (2, 8), and (3, 21), provide a concise representation of the function's behavior over this domain. They can be used to plot the function on a graph, to analyze its rate of change, or for other mathematical purposes. The process of function evaluation has been consistently applied across all three values in the domain, demonstrating the reliability and accuracy of the method. The careful application of the order of operations is crucial for obtaining correct results, and this example illustrates the step-by-step process involved in evaluating a function for a given domain. By understanding and mastering function evaluation, you can gain a deeper insight into the behavior of mathematical functions and their applications in various fields.
Conclusion
In conclusion, we have successfully evaluated the function f(x) = 3x² - 2x for the domain 1, 2, 3}. This function evaluation process involved substituting each value from the domain into the function's expression and simplifying the resulting expression using the order of operations. We found that f(1) = 1, f(2) = 8, and f(3) = 21. These results provide a clear understanding of the function's behavior over the specified domain and illustrate the fundamental principles of function evaluation. The process of function evaluation is a cornerstone of mathematical analysis. It allows us to determine the output of a function for specific input values, providing valuable insights into the function's characteristics and behavior. This process is essential for graphing functions, solving equations, modeling real-world phenomena, and numerous other applications in mathematics and related fields. Understanding the domain of a function is crucial for determining the valid input values. In this case, the domain was explicitly given as {1, 2, 3}, which simplified the evaluation process. However, for other functions, the domain may be implied by the function's expression, and it is important to identify any restrictions on the input values, such as division by zero or taking the square root of a negative number. The order of operations (PEMDAS/BODMAS) plays a critical role in accurate function evaluation. By following the correct order, we can ensure that the expression is simplified correctly, leading to the correct output value. This involves first evaluating exponents, then performing multiplications and divisions from left to right, and finally performing additions and subtractions from left to right. The results of our evaluation, f(1) = 1, f(2) = 8, and f(3) = 21, provide us with three ordered pairs, demonstrating the fundamental principles and techniques of function evaluation. This process is a crucial skill in mathematics and has wide-ranging applications in various fields. By understanding and mastering function evaluation, you can gain a deeper appreciation for the power and versatility of mathematical functions.
