Evaluating Piecewise Functions A Step-by-Step Guide

Jul 10, 2025 by ADMIN 52 views

$Evaluating the Piecewise Function f(x) = {-3x + 9 if x ≤ -8, 5 if -8 < x ≤ 2, x^2 - 3 if x > 2}$

In the realm of mathematics, piecewise functions stand out as versatile tools for modeling situations where different rules apply over different intervals. These functions, defined by multiple sub-functions each applicable to a specific domain, are prevalent in various real-world applications, ranging from tax brackets to signal processing. Understanding how to evaluate piecewise functions is crucial for anyone seeking to grasp their intricacies and applications. In this comprehensive guide, we will delve into the evaluation of a specific piecewise function, $f(x)$ , defined as follows:

$f(x)=\left\{\begin{array}{ll}-3 x+9 & \text { if } x \leq-8 \\5 & \text { if }-8 < x \leq 2 \\x^2 - 3 & \text { if } x > 2\end{array}\right.$

This function comprises three distinct sub-functions, each governing a specific interval of the input variable $x$ . To evaluate $f(x)$ for a given value of $x$ , we must first determine which interval $x$ falls into and then apply the corresponding sub-function. Let's embark on a step-by-step journey to master the art of evaluating this piecewise function.

Unveiling the Sub-Functions: A Detailed Exploration

Before we delve into the evaluation process, let's take a closer look at the three sub-functions that constitute our piecewise function $f(x)$ .

Sub-function 1: -3x + 9 (if x ≤ -8)

This sub-function, defined as $-3x + 9$ , applies when the input value $x$ is less than or equal to -8. It represents a linear function with a slope of -3 and a y-intercept of 9. This means that as $x$ decreases, the value of the function increases, and when $x$ is -8, the function value is -3(-8) + 9 = 33.
Sub-function 2: 5 (if -8 < x ≤ 2)

This sub-function, simply the constant value 5, applies when $x$ is greater than -8 but less than or equal to 2. This indicates that for any $x$ within this interval, the function's output will always be 5. This sub-function represents a horizontal line at y = 5.
Sub-function 3: x^2 - 3 (if x > 2)

This sub-function, defined as $x^2 - 3$ , applies when $x$ is strictly greater than 2. It represents a quadratic function, specifically a parabola shifted 3 units downward. As $x$ increases beyond 2, the function's value increases rapidly due to the squared term. When $x$ is 3, the function value is 3^2 - 3 = 6.

Understanding the behavior of each sub-function within its respective interval is crucial for accurately evaluating the piecewise function for different input values.

Step-by-Step Evaluation: A Practical Guide

Now that we have a firm grasp of the sub-functions, let's outline the step-by-step process for evaluating the piecewise function $f(x)$ for a given input value $x$ .

Identify the Interval: The first step is to determine which interval the input value $x$ belongs to. This involves comparing $x$ to the boundary values that define the intervals: -8 and 2. Ask yourself: Is $x$ less than or equal to -8? Is $x$ greater than -8 but less than or equal to 2? Or is $x$ strictly greater than 2?
Select the Corresponding Sub-function: Once the interval is identified, select the sub-function that corresponds to that interval. For instance, if $x$ is less than or equal to -8, choose the sub-function $-3x + 9$ . If $x$ falls between -8 and 2, select the sub-function 5. And if $x$ is greater than 2, opt for the sub-function $x^2 - 3$ .
Substitute and Evaluate: After selecting the appropriate sub-function, substitute the input value $x$ into the sub-function's expression. Then, perform the necessary calculations to evaluate the expression. The result of this evaluation is the value of the piecewise function $f(x)$ for the given input $x$ .

Let's illustrate this process with a few examples to solidify your understanding.

Practical Examples: Putting the Steps into Action

To further enhance your comprehension, let's work through several examples of evaluating the piecewise function $f(x)$ for different input values.

(a) Evaluating f(7)

Identify the Interval: First, we need to determine which interval 7 belongs to. Since 7 is greater than 2, it falls into the interval $x > 2$ .
Select the Corresponding Sub-function: Based on the interval, we select the sub-function that applies when $x > 2$ , which is $x^2 - 3$ .
Substitute and Evaluate: Now, we substitute $x = 7$ into the sub-function: $f(7) = 7^2 - 3 = 49 - 3 = 46$ .

Therefore, $f(7) = 46$ .

(b) Evaluating f(-10)

Identify the Interval: We need to find the interval that -10 belongs to. Since -10 is less than or equal to -8, it falls into the interval $x \leq -8$ .
Select the Corresponding Sub-function: For the interval $x \leq -8$ , we select the sub-function $-3x + 9$ .
Substitute and Evaluate: Substituting $x = -10$ into the sub-function, we get: $f(-10) = -3(-10) + 9 = 30 + 9 = 39$ .

Therefore, $f(-10) = 39$ .

(c) Evaluating f(-5.5)

Identify the Interval: We determine which interval -5.5 belongs to. Since -5.5 is greater than -8 but less than or equal to 2, it falls into the interval $-8 < x \leq 2$ .
Select the Corresponding Sub-function: For this interval, we select the sub-function 5.
Substitute and Evaluate: Since the sub-function is a constant value, there's no need to substitute. $f(-5.5) = 5$ .

Therefore, $f(-5.5) = 5$ .

(d) Evaluating f(-10.9)

Identify the Interval: We find the interval that -10.9 belongs to. Since -10.9 is less than or equal to -8, it falls into the interval $x \leq -8$ .
Select the Corresponding Sub-function: For the interval $x \leq -8$ , we select the sub-function $-3x + 9$ .
Substitute and Evaluate: Substituting $x = -10.9$ into the sub-function, we get: $f(-10.9) = -3(-10.9) + 9 = 32.7 + 9 = 41.7$ .

Therefore, $f(-10.9) = 41.7$ .

Summary of Results

To summarize our evaluations:

(a) f(7) = 46
(b) f(-10) = 39
(c) f(-5.5) = 5
(d) f(-10.9) = 41.7

These examples demonstrate the systematic approach to evaluating piecewise functions. By carefully identifying the interval and applying the corresponding sub-function, we can accurately determine the function's value for any given input.

Key Takeaways and Further Exploration

Mastering the evaluation of piecewise functions opens doors to a deeper understanding of mathematical modeling and its applications in various fields. Here are some key takeaways to solidify your knowledge:

Piecewise functions are defined by multiple sub-functions, each applicable to a specific domain.
To evaluate a piecewise function, identify the interval the input belongs to and apply the corresponding sub-function.
Understanding the behavior of each sub-function is crucial for accurate evaluation.
Piecewise functions are widely used in real-world applications, such as tax brackets and signal processing.

To further expand your knowledge, consider exploring the following topics:

Graphing piecewise functions: Visualizing piecewise functions can provide valuable insights into their behavior.
Applications of piecewise functions: Investigate how piecewise functions are used in various fields, such as economics and engineering.
Continuity and differentiability of piecewise functions: Explore the conditions under which piecewise functions are continuous and differentiable.

By continuing your exploration, you'll unlock the full potential of piecewise functions and their applications in the mathematical world and beyond. Remember, practice makes perfect, so continue working through examples and challenging yourself to deepen your understanding.