Evaluating Piecewise Functions Finding F(3) Explained

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In the realm of mathematics, piecewise functions hold a significant role in modeling real-world scenarios that exhibit varying behavior across different intervals. These functions are defined by multiple sub-functions, each applicable over a specific domain. To truly grasp the essence of piecewise functions, it's essential to understand how to evaluate them at given points, particularly at the boundaries where the sub-functions transition. In this comprehensive exploration, we will delve into the intricacies of evaluating a specific piecewise function at x = 3, providing a detailed justification for the obtained value. This article aims to provide a deep understanding of piecewise functions and their evaluation, targeting students, educators, and anyone interested in mathematical concepts.

The function we'll be focusing on is defined as follows:

f(x) = { 2x - 3,  x > 3
        { -x^2 + 15, x <= 3

Our primary goal is to determine the value of f(3) and provide a clear, step-by-step justification for our answer. This exploration will not only solidify your understanding of piecewise function evaluation but also enhance your problem-solving skills in mathematics.

Defining Piecewise Functions

To effectively evaluate piecewise functions, it is crucial to first understand their fundamental structure. A piecewise function, by definition, is a function that is defined by multiple sub-functions, each of which applies to a specific interval of the domain. These intervals are often defined using inequalities, and the function's overall behavior is determined by which interval the input value falls into. The main keyword here is understanding the conditions for each piece of the function.

Consider the piecewise function we are examining:

f(x) = { 2x - 3,  x > 3
        { -x^2 + 15, x <= 3

This function has two sub-functions:

  1. 2x - 3: This sub-function is applicable when x is strictly greater than 3 (x > 3). This means that for any value of x that is larger than 3, we will use this expression to calculate the function's value.
  2. -x² + 15: This sub-function is applicable when x is less than or equal to 3 (x ≤ 3). Therefore, for any value of x that is 3 or smaller, we will use this expression to calculate the function's value.

The critical aspect of piecewise functions lies in recognizing which sub-function to use for a given input value. The inequalities associated with each sub-function serve as the guiding rules, dictating the function's behavior across different parts of its domain. Understanding these conditions is essential for accurate evaluation and analysis of piecewise functions.

Evaluating f(3): A Step-by-Step Approach

Now, let's focus on evaluating f(3) for the given piecewise function. This involves carefully determining which sub-function applies when x = 3. The key to evaluating piecewise functions is to identify the correct interval for the input value.

Recall the function definition:

f(x) = { 2x - 3,  x > 3
        { -x^2 + 15, x <= 3

We are interested in finding f(3), which means we need to determine which sub-function applies when x is equal to 3. Examining the conditions associated with each sub-function:

  • The first sub-function, 2x - 3, is defined for x > 3.
  • The second sub-function, -x² + 15, is defined for x ≤ 3.

Since 3 is less than or equal to 3 (3 ≤ 3), the second sub-function, -x² + 15, is the one that applies in this case. This is a crucial step, as choosing the correct sub-function is paramount to obtaining the correct result. It's important to pay close attention to the inequality signs to ensure the appropriate sub-function is selected.

Now that we have identified the correct sub-function, we can proceed with the evaluation. We substitute x = 3 into the expression -x² + 15:

f(3) = -(3)² + 15

Following the order of operations (PEMDAS/BODMAS), we first calculate the exponent:

f(3) = -9 + 15

Then, we perform the addition:

f(3) = 6

Therefore, the value of f(3) for the given piecewise function is 6. This step-by-step evaluation demonstrates the importance of carefully selecting the appropriate sub-function based on the given input value.

Justifying the Answer: Why -x² + 15?

The justification for using the sub-function -x² + 15 when evaluating f(3) lies directly in the definition of the piecewise function. Piecewise functions, by their nature, are defined differently across different intervals of their domain. The justification is rooted in the condition associated with each sub-function.

Let's reiterate the function definition:

f(x) = { 2x - 3,  x > 3
        { -x^2 + 15, x <= 3

As we established earlier, the function f(x) behaves according to the following rules:

  • When x is strictly greater than 3 (x > 3), the function is defined by the expression 2x - 3.
  • When x is less than or equal to 3 (x ≤ 3), the function is defined by the expression -x² + 15.

Our task was to find f(3), which means we needed to determine the function's value when x is exactly 3. Examining the conditions, we see that the inequality x ≤ 3 includes the case where x is equal to 3. The key point is the presence of the