Evaluating Piecewise Function Exercise B.1 Explained

Jul 4, 2025 by Admin 53 views

Exercise B.1 Evaluating a Piecewise Function

In mathematics, a piecewise function is a function defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Piecewise functions are very important tools in modeling real-world phenomena, especially when a single formula can't accurately describe the function's behavior across its entire input range. These functions are defined by different formulas or expressions for different intervals of their domain. Understanding how to evaluate piecewise functions is a fundamental skill in algebra and calculus. This article will delve into the process of evaluating a specific piecewise function, providing a step-by-step guide to ensure clarity and comprehension.

This article aims to provide a comprehensive explanation of how to evaluate a piecewise function, walking through each step with detailed examples. By understanding the mechanics of piecewise functions, readers will be better equipped to tackle more complex mathematical problems and real-world applications. This introduction sets the stage for a detailed exploration of piecewise functions, emphasizing their importance and relevance in various mathematical contexts.

Before we dive into evaluating the function, let's define it clearly. The function we're working with is:

f(x) = 
  \begin{cases}
    x^2 + 2x, & \text{if } x \leq -1 \\
    x, & \text{if } -1 < x \leq 1 \\
    -1, & \text{if } x > 1
  \end{cases}

This piecewise function, denoted as f(x), consists of three sub-functions, each defined over a specific interval of the input x. The first sub-function, x² + 2x, applies when x is less than or equal to -1. The second sub-function, x, applies when x is strictly greater than -1 but less than or equal to 1. The third sub-function, -1, applies when x is strictly greater than 1. Understanding these intervals is crucial for correctly evaluating the function for different values of x.

The function's definition is segmented into three distinct rules, each governing a specific portion of the domain. It is essential to understand each part of the definition to evaluate the function correctly. For x values less than or equal to -1, we use the quadratic expression x² + 2x. For x values strictly between -1 and 1 (inclusive of 1), the function simplifies to x itself. Lastly, for x values greater than 1, the function is a constant value of -1. This structure allows the function to behave differently across its domain, providing flexibility in modeling diverse mathematical scenarios.

Now, let's evaluate the function for the given values:

a. f(-1)

To find f(-1), we first need to determine which interval -1 falls into. Looking at the function definition, -1 satisfies the condition x ≤ -1. Therefore, we use the first sub-function:

f(-1) = (-1)² + 2(-1) = 1 - 2 = -1

Thus, f(-1) = -1. This first evaluation demonstrates the crucial step of identifying the correct interval before applying the corresponding sub-function. By correctly identifying that -1 falls into the interval x ≤ -1, we are able to use the appropriate expression, x² + 2x, to find the value of the function. This process highlights the piecewise nature of the function, where the value depends on the specific interval in which the input lies. The calculation itself involves basic arithmetic operations, making it straightforward once the correct sub-function is chosen.

b. f(5)

For f(5), we observe that 5 is greater than 1, so we use the third sub-function, which is a constant:

f(5) = -1

Therefore, f(5) = -1. This case illustrates how, for values in certain intervals, a piecewise function can have a constant output. The simplicity of this evaluation underscores the importance of recognizing which part of the function's definition applies to the given input. When x is greater than 1, the function f(x) is defined as -1, making the evaluation direct and unambiguous. This behavior is typical of piecewise functions, where certain intervals may result in constant, easily determinable values.

c. f(3/4)

To evaluate f(3/4), we see that 3/4 falls between -1 and 1, specifically in the interval -1 < x ≤ 1. Therefore, we use the second sub-function:

f(3/4) = 3/4

Hence, f(3/4) = 3/4. This evaluation demonstrates a straightforward application of one of the sub-functions. Since 3/4 lies within the interval where f(x) = x, the function's value is simply the input itself. This highlights the piecewise function's ability to define different behaviors across its domain, with the second sub-function acting as an identity function within its specific interval. Recognizing that 3/4 falls within the specified range allows for a direct substitution and a simple result.

d. f(0)

For f(0), 0 also falls in the interval -1 < x ≤ 1, so we use the second sub-function:

f(0) = 0

Thus, f(0) = 0. This example further reinforces the concept of direct evaluation within a specified interval. The value of f(x) is simply x for inputs between -1 and 1, making the evaluation at 0 a trivial substitution. This emphasizes the importance of correctly identifying the interval to which the input belongs, as it dictates which sub-function should be used. The simplicity of this case helps solidify understanding of the piecewise function's behavior within its defined segments.

e. f(-1/2)

For f(-1/2), -1/2 lies between -1 and 1, so we again use the second sub-function:

f(-1/2) = -1/2

Therefore, f(-1/2) = -1/2. This final evaluation reinforces the principle that inputs within the interval -1 < x ≤ 1 are directly mapped to their respective values by the piecewise function. The straightforward substitution confirms the function's behavior within this segment, further illustrating the importance of understanding the intervals and their corresponding sub-functions. By consistently applying this approach, one can accurately evaluate piecewise functions for any given input.

Here's a summary of our evaluations:

f(-1) = -1
f(5) = -1
f(3/4) = 3/4
f(0) = 0
f(-1/2) = -1/2

This summary provides a concise overview of the results obtained for each input value. Each evaluation underscores the importance of identifying the correct interval in the piecewise function's definition before applying the corresponding sub-function. The variety of results, from constant values to direct substitutions, demonstrates the flexibility and utility of piecewise functions in modeling different scenarios. This compilation serves as a quick reference and reinforces the key concepts covered in the evaluation process.

Evaluating a piecewise function involves determining which interval the input value belongs to and then applying the corresponding sub-function. This process requires careful attention to the function's definition and the conditions for each interval. Through the examples provided, we've demonstrated how to evaluate a specific piecewise function for various input values. Understanding this process is essential for working with more complex mathematical functions and applications.

The process of evaluating piecewise functions is a fundamental skill in mathematics, with applications spanning various fields, including engineering, computer science, and economics. Piecewise functions allow for the modeling of phenomena that exhibit different behaviors under different conditions. By mastering the techniques discussed in this article, readers can confidently approach and solve problems involving piecewise functions. The ability to correctly evaluate these functions is a stepping stone towards more advanced mathematical concepts and practical problem-solving.

To further solidify your understanding, try evaluating the following piecewise functions for different values of x:

g(x) = { x + 1, if x < 0; x^2, if x ≥ 0 }
h(x) = { -x, if x < -2; 3, if -2 ≤ x ≤ 2; x, if x > 2 }

These exercises provide an opportunity to apply the concepts and methods discussed in this article. By working through these problems, readers can reinforce their understanding of piecewise functions and improve their problem-solving skills. Practice is crucial in mastering mathematical concepts, and these exercises are designed to offer a practical application of the evaluation process. Engaging with these exercises will help solidify your grasp of how piecewise functions operate and how to evaluate them effectively.