Function Mapping Find The Function Where X=2 Maps To 32

In mathematics, functions are essential tools for describing relationships between variables. Understanding how a function maps an input value to an output value is fundamental. This article will explore the process of determining which function, among a given set, maps the input x = 2 to the output 32. We will systematically evaluate each function, providing a detailed explanation of the substitution and calculation steps. This will not only help in solving the specific problem but also strengthen your overall understanding of function evaluation and analysis. Understanding functions and their mappings is a core concept in algebra and calculus, crucial for solving various mathematical problems and real-world applications. Before diving into the solution, let's define what a function is and how it operates. A function is a rule that assigns to each input value exactly one output value. This input is often denoted by x, and the output is represented by f(x), g(x), h(x), and so on, depending on the function's name. The process of finding the output for a given input is called function evaluation. In this case, we want to find the function that satisfies the condition f(2) = 32, g(2) = 32, h(2) = 32, or j(2) = 32. By substituting x = 2 into each function and simplifying, we can identify the correct function. This method is straightforward but requires careful arithmetic and attention to detail. Now, let's proceed with the problem and identify the function that maps x = 2 to 32.

Evaluating the Functions

To determine which function maps x = 2 to 32, we need to evaluate each function at x = 2 and check the resulting output. This involves substituting x = 2 into each function's expression and simplifying the expression using the order of operations. This systematic approach ensures that we accurately identify the function that satisfies the given condition. Let's begin with the first function and proceed through each one step by step. This methodical approach not only solves the problem but also reinforces the process of function evaluation, a crucial skill in mathematics. By following this process, we can confidently identify the function that meets the specified criteria. Accurate substitution and simplification are the keys to successfully evaluating functions. Each function presents a slightly different algebraic expression, requiring us to apply the order of operations carefully. This practice is essential for mastering algebraic manipulations and understanding how functions behave. Let's start by evaluating the first function, f(x), at x = 2.

Evaluating f(x) = -3x² - 4

To evaluate the function f(x) = -3x² - 4 at x = 2, we substitute 2 for x in the expression. This gives us f(2) = -3(2)² - 4. The next step is to follow the order of operations, which dictates that we first handle the exponent. So, we calculate 2², which is 4. Now our expression becomes f(2) = -3(4) - 4. Next, we perform the multiplication: -3 multiplied by 4 equals -12. So, we have f(2) = -12 - 4. Finally, we perform the subtraction: -12 minus 4 equals -16. Therefore, f(2) = -16. Since -16 is not equal to 32, the function f(x) = -3x² - 4 does not map x = 2 to 32. This process demonstrates the importance of following the order of operations to arrive at the correct result. Each step must be performed in the correct sequence to ensure accuracy. Function evaluation is a fundamental skill in algebra, and this example illustrates the process clearly. The result, f(2) = -16, helps us eliminate this function as a potential solution. Now, let's move on to the next function and repeat the process to see if it maps x = 2 to 32. This systematic approach will help us identify the correct function efficiently.

Evaluating g(x) = 4(x + 3)² - 68

Now, let's evaluate the function g(x) = 4(x + 3)² - 68 at x = 2. We substitute 2 for x in the expression, resulting in g(2) = 4(2 + 3)² - 68. Following the order of operations, we first address the parentheses: 2 + 3 equals 5. The expression now becomes g(2) = 4(5)² - 68. Next, we handle the exponent: 5² is 25. So, the expression is g(2) = 4(25) - 68. Now, we perform the multiplication: 4 multiplied by 25 equals 100. The expression is now g(2) = 100 - 68. Finally, we perform the subtraction: 100 minus 68 equals 32. Thus, g(2) = 32. Since g(2) equals 32, we have found the function that maps x = 2 to 32. This function evaluation involved several steps, each requiring careful attention to detail. The successful evaluation demonstrates the importance of following the order of operations and accurately performing each arithmetic step. This process not only solves the problem but also reinforces the concept of function evaluation and algebraic manipulation. The result, g(2) = 32, confirms that g(x) is the function we were looking for. However, to ensure we have thoroughly explored all possibilities and to further illustrate the process, let's continue by evaluating the remaining functions. This will provide additional practice and reinforce our understanding of function evaluation.

Evaluating h(x) = 3x

To evaluate the function h(x) = 3x at x = 2, we substitute 2 for x in the expression. This gives us h(2) = 3(2). Next, we perform the multiplication: 3 multiplied by 2 equals 6. Therefore, h(2) = 6. Since 6 is not equal to 32, the function h(x) = 3x does not map x = 2 to 32. This evaluation is straightforward and highlights the basic process of function substitution. The result, h(2) = 6, eliminates this function as a possible solution. This simple example reinforces the concept of function evaluation and demonstrates how different functions behave with the same input value. Now, let's proceed to the final function to complete our analysis. Evaluating the last function will provide further practice and ensure we have considered all options. This thorough approach helps to solidify our understanding of function evaluation and problem-solving techniques.

Evaluating j(x) = 2x - 62

Finally, let's evaluate the function j(x) = 2x - 62 at x = 2. We substitute 2 for x in the expression, giving us j(2) = 2(2) - 62. Following the order of operations, we first perform the multiplication: 2 multiplied by 2 equals 4. So, the expression becomes j(2) = 4 - 62. Next, we perform the subtraction: 4 minus 62 equals -58. Therefore, j(2) = -58. Since -58 is not equal to 32, the function j(x) = 2x - 62 does not map x = 2 to 32. This evaluation completes our analysis of all the given functions. The result, j(2) = -58, further reinforces the process of function evaluation and confirms that not all functions will produce the desired output for a given input. This comprehensive approach ensures that we have thoroughly examined each function and have accurately identified the one that satisfies the given condition. The systematic evaluation of each function has provided valuable practice in algebraic manipulation and function analysis.

Conclusion

In conclusion, by evaluating each function at x = 2, we found that the function g(x) = 4(x + 3)² - 68 maps x = 2 to 32. The other functions, f(x) = -3x² - 4, h(x) = 3x, and j(x) = 2x - 62, did not produce the output of 32 when x = 2. This exercise demonstrates the importance of systematic function evaluation and accurate application of the order of operations. Understanding how to evaluate functions is a fundamental skill in mathematics, with applications in various fields such as algebra, calculus, and computer science. This process involves substituting the given input value into the function's expression and simplifying to obtain the output. Each step, from substitution to simplification, requires careful attention to detail to ensure accuracy. The ability to correctly evaluate functions is essential for solving mathematical problems, modeling real-world phenomena, and understanding complex relationships between variables. The methodical approach we used in this article can be applied to evaluate any function at any given point. By following the order of operations and paying close attention to arithmetic, we can confidently determine the output of a function for a specific input. This skill is crucial for further studies in mathematics and related disciplines.