Identifying Functions Mapping X=2 To 32
In the realm of mathematics, functions serve as fundamental tools for describing relationships between variables. A function essentially acts as a mapping mechanism, transforming an input value (often denoted as x) into a corresponding output value. Understanding how different functions behave and how specific inputs are mapped to outputs is crucial for various mathematical applications.
This article delves into the process of identifying functions that map the input x = 2 to the output 32. We will explore four distinct functions, each with its own unique characteristics, and systematically evaluate them to determine if they satisfy the given condition. By carefully analyzing each function, we aim to enhance your understanding of function evaluation and deepen your ability to work with mathematical expressions.
The problem at hand presents us with four functions: f(x), g(x), h(x), and j(x). Our goal is to determine which of these functions, when evaluated at x = 2, yields an output of 32. This involves substituting 2 for x in each function and simplifying the resulting expression. Let's embark on this journey of mathematical exploration and unravel the mysteries of function mapping.
Function Evaluation: A Step-by-Step Approach
Evaluating functions is a core skill in mathematics, and it involves substituting a given value for the variable in the function's expression and then simplifying the result. This process allows us to determine the output of the function for a specific input, providing valuable insights into the function's behavior. To master function evaluation, a systematic approach is key. Let's break down the process into manageable steps.
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Identify the function: Begin by clearly identifying the function you want to evaluate. This includes understanding the function's name and its algebraic expression. For instance, in our case, we have functions like f(x) = -3x^2 - 4 and g(x) = 4(x + 3)^2 - 68. Knowing the function's expression is the foundation for the subsequent steps.
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Substitute the input value: The next step is to substitute the given input value for the variable (x in most cases) in the function's expression. This means replacing every instance of x with the specified value. For example, if we want to evaluate f(x) at x = 2, we would replace x with 2 in the expression, resulting in f(2) = -3(2)^2 - 4.
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Simplify the expression: Once the substitution is done, the next crucial step is to simplify the resulting expression. This involves performing the necessary arithmetic operations, following the order of operations (PEMDAS/BODMAS). This often includes dealing with exponents, multiplication, division, addition, and subtraction. Continuing our example, f(2) = -3(2)^2 - 4 simplifies to f(2) = -3(4) - 4, then to f(2) = -12 - 4, and finally to f(2) = -16.
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Determine the output: After simplifying the expression, the final step is to determine the output value. This is the value that the function maps the input to. In our example, the output of f(x) when x = 2 is -16. This means that the function f(x) maps the input 2 to the output -16.
By following these steps diligently, you can confidently evaluate functions for any given input. This skill is fundamental for various mathematical tasks, including graphing functions, solving equations, and modeling real-world phenomena.
Evaluating the Given Functions
Now, let's apply our function evaluation skills to the four functions provided in the problem. We will systematically evaluate each function at x = 2 and determine which one(s) produce an output of 32.
Evaluating f(x) = -3x^2 - 4
To evaluate f(x) at x = 2, we substitute 2 for x in the function's expression:
f(2) = -3(2)^2 - 4
Next, we simplify the expression following the order of operations:
f(2) = -3(4) - 4
f(2) = -12 - 4
f(2) = -16
Therefore, f(2) = -16. This means that the function f(x) maps the input 2 to the output -16, which is not equal to 32. So, f(x) is not the function we are looking for.
Evaluating g(x) = 4(x + 3)^2 - 68
Now, let's evaluate g(x) at x = 2. We substitute 2 for x in the function's expression:
g(2) = 4(2 + 3)^2 - 68
We simplify the expression, following the order of operations:
g(2) = 4(5)^2 - 68
g(2) = 4(25) - 68
g(2) = 100 - 68
g(2) = 32
Therefore, g(2) = 32. This means that the function g(x) maps the input 2 to the output 32. Thus, g(x) is one of the functions that satisfies our condition.
Evaluating h(x) = 3x
Let's proceed to evaluate h(x) at x = 2. We substitute 2 for x in the function's expression:
h(2) = 3(2)
Simplifying the expression gives us:
h(2) = 6
Therefore, h(2) = 6. This indicates that the function h(x) maps the input 2 to the output 6, which is not equal to 32. Consequently, h(x) is not the function we seek.
Evaluating j(x) = 2x - 62
Finally, we evaluate j(x) at x = 2. We substitute 2 for x in the function's expression:
j(2) = 2(2) - 62
Simplifying the expression yields:
j(2) = 4 - 62
j(2) = -58
Therefore, j(2) = -58. This demonstrates that the function j(x) maps the input 2 to the output -58, which is not equal to 32. Hence, j(x) is not the function we are looking for.
Conclusion: Identifying the Function
Through our systematic evaluation of the four functions, we have discovered that only one function, g(x) = 4(x + 3)^2 - 68, maps the input x = 2 to the output 32. The other functions, f(x), h(x), and j(x), produce different output values when evaluated at x = 2.
This exercise highlights the importance of function evaluation in determining the relationship between inputs and outputs. By carefully substituting and simplifying, we can gain valuable insights into the behavior of functions and their applications in various mathematical contexts. Understanding function mapping is crucial for solving equations, modeling real-world phenomena, and making predictions based on mathematical relationships.
In summary, the function that maps x = 2 to 32 is g(x) = 4(x + 3)^2 - 68.
This exploration not only answers the specific question posed but also reinforces the fundamental concepts of function evaluation and mapping, which are essential for a strong foundation in mathematics.
