Piecewise Function Evaluation A Step By Step Guide To Finding F 3

Jun 24, 2025 by ADMIN 66 views

Given the piecewise function $f(x)=\left{\begin{array}{cc}-x, & x \leq-1 \\ 1, & x=0 \\ x+1 & x \geq 1\end{array}\right.$, what is the value of $f(3)$?

Piecewise Function Evaluation: Finding f(3)

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In the realm of mathematics, piecewise functions stand out as fascinating entities. They're like chameleons, changing their behavior based on the input value. This article delves into the world of piecewise functions, focusing on how to evaluate them for specific inputs. We'll tackle a specific example to illustrate the process, providing a clear and comprehensive understanding of how these functions operate.

Understanding Piecewise Functions

At its core, a piecewise function is a function defined by multiple sub-functions, each applicable over a certain interval of the domain. Think of it as a function with different rules for different 'neighborhoods' of input values. These 'neighborhoods' are defined by specific conditions or intervals, creating distinct segments within the function's domain. Each sub-function has its own formula, and the appropriate formula is used depending on which interval the input value falls into.

The beauty of piecewise functions lies in their ability to model situations where the relationship between input and output changes abruptly or follows different patterns across different ranges. From tax brackets to step functions, piecewise functions are powerful tools for representing real-world scenarios with varying conditions. To truly grasp their power, let's delve deeper into the structure and evaluation process.

Let's consider the general structure of a piecewise function. Typically, it's represented using a notation that lists each sub-function along with its corresponding domain interval. For example:

f(x) = {
  sub-function_1, condition_1
  sub-function_2, condition_2
  ...
  sub-function_n, condition_n
}

Here, sub-function_i is the formula for the i-th sub-function, and condition_i specifies the interval of x-values for which that sub-function applies. The conditions are crucial, as they dictate which sub-function to use for a given input. The conditions must be mutually exclusive, ensuring that for any input value, only one sub-function applies. This prevents ambiguity and guarantees a well-defined function.

Now, let's talk about evaluating piecewise functions. The process is straightforward: given an input value, we first identify the interval to which it belongs based on the conditions. Then, we apply the corresponding sub-function's formula to the input value. It's like following a set of instructions – each instruction applies only in a specific situation. This conditional application of sub-functions is what gives piecewise functions their unique character and flexibility.

For instance, if we have a piecewise function defined with three sub-functions and we want to find the value of the function at a particular x, we first check which of the three conditions x satisfies. Once we've identified the correct condition, we use the corresponding sub-function to calculate the function's value. This step-by-step approach ensures that we're using the appropriate formula for the given input, leading to an accurate result. The key is to pay close attention to the conditions and match the input value to the correct interval.

The Given Piecewise Function

Now, let's turn our attention to the specific piecewise function we'll be working with:

$f(x)=\left{\begin{array}{cc}-x, & x \leq-1 \\ 1, & x=0 \\ x+1 & x \geq 1\end{array}\right.$

This function, denoted as f(x), is composed of three distinct sub-functions, each defined over a specific range of x values. The first sub-function, -x, is active when x is less than or equal to -1. The second sub-function is a constant, 1, and it's specifically defined for x equals 0. The third sub-function, x+1, comes into play when x is greater than or equal to 1. Notice the clear separation of domains, ensuring that there's no overlap or ambiguity in the function's definition. The function's behavior is distinct and well-defined across its entire domain.

To fully appreciate this piecewise function, let's break down each sub-function and its corresponding domain.

Sub-function 1: -x for x ≤ -1 This sub-function is a simple linear function that reflects the input value across the y-axis. For any x value less than or equal to -1, the function returns the negative of that value. For example, if x is -2, then f(x) would be -(-2) = 2. This sub-function defines the behavior of the piecewise function for all x values in the interval (-∞, -1].
Sub-function 2: 1 for x = 0 This sub-function is a constant function, meaning it always returns the same value regardless of the input, as long as the input is exactly 0. In this case, when x is 0, f(x) is always 1. This is a single point on the graph of the piecewise function, and it's crucial to include it as it defines the function's value at x = 0.
Sub-function 3: x + 1 for x ≥ 1 This sub-function is another linear function, but this time it adds 1 to the input value. For any x value greater than or equal to 1, the function returns x plus 1. For instance, if x is 2, then f(x) would be 2 + 1 = 3. This sub-function defines the behavior of the piecewise function for all x values in the interval [1, ∞).

Understanding these sub-functions and their domains is the first step in evaluating the piecewise function for any given input. We must carefully consider which condition the input satisfies to determine which sub-function to apply. This careful consideration ensures we arrive at the correct output value.

Evaluating f(3)

Our mission now is to determine the value of f(x) when x equals 3. To do this, we must carefully navigate the different segments of our piecewise function and identify the correct sub-function to apply. The key is to match the given x value with the appropriate condition specified in the function's definition. Let's walk through the process step by step.

First, let's revisit the piecewise function:

$f(x)=\left{\begin{array}{cc}-x, & x \leq-1 \\ 1, & x=0 \\ x+1 & x \geq 1\end{array}\right.$

We need to find the value of f(3). The crucial step is to determine which condition x = 3 satisfies. We examine each condition:

x ≤ -1: Is 3 less than or equal to -1? No. This condition does not apply.
x = 0: Is 3 equal to 0? No. This condition also does not apply.
x ≥ 1: Is 3 greater than or equal to 1? Yes! This condition is satisfied.

Since x = 3 satisfies the condition x ≥ 1, we know that we should use the third sub-function to evaluate f(3). The third sub-function is:

f(x) = x + 1 for x ≥ 1

Now, we simply substitute x = 3 into this sub-function:

f(3) = 3 + 1

Performing the addition, we get:

f(3) = 4

Therefore, the value of the piecewise function f(x) when x is 3 is 4. This result highlights the importance of carefully considering the conditions when evaluating piecewise functions. By correctly identifying the relevant condition, we ensure that we apply the appropriate sub-function and arrive at the accurate output value.

The Answer

Having meticulously evaluated our piecewise function, we've arrived at the answer. The value of f(x) when x = 3 is 4. Therefore, the correct answer is:

D. 4

This exercise underscores the core principle of working with piecewise functions: carefully matching the input value with the correct condition to select the appropriate sub-function. This step-by-step approach guarantees accurate evaluation and a clear understanding of the function's behavior across its domain. This detailed solution serves as a valuable example for tackling similar problems and solidifies the understanding of piecewise functions.

Conclusion

In this exploration of piecewise functions, we've uncovered the method for evaluating them at specific points. By carefully examining the conditions that define each sub-function's domain, we can confidently determine which formula to apply for a given input. Our example, finding f(3) for the given piecewise function, demonstrates this process in action, leading us to the correct answer of 4.

Piecewise functions, with their segmented definitions, are powerful tools in mathematics and various applications. They allow us to model situations where the relationship between variables changes depending on the input value. Understanding how to evaluate these functions is crucial for anyone working with mathematical models and real-world problems. The ability to navigate the different sub-functions and their domains opens up a world of possibilities for representing and analyzing complex relationships.

By mastering the techniques discussed in this article, you'll be well-equipped to tackle a wide range of problems involving piecewise functions. From determining the output for a specific input to graphing the function's behavior, the principles remain the same: understand the conditions, select the appropriate sub-function, and apply the formula. With practice and a solid understanding of the underlying concepts, you'll be able to confidently navigate the fascinating world of piecewise functions and their applications.