Evaluating Piecewise Functions Finding F(3) Explained

Evaluating piecewise functions requires careful consideration of the input value and the domain intervals defined for each function piece. This article focuses on how to accurately determine the value of a piecewise function at a specific point, using the example of ${ f(3) }$ for the given function. Piecewise functions, which are defined by multiple sub-functions each applicable over a certain interval of the domain, are a fundamental concept in mathematics. Mastering the evaluation of these functions is crucial for various mathematical applications and problem-solving scenarios. In this comprehensive guide, we will delve into the step-by-step process of evaluating ${ f(3) }$ for the provided piecewise function, highlighting the importance of correctly identifying the relevant domain interval and applying the corresponding function rule. Understanding piecewise functions is not only essential for academic pursuits but also for real-world applications where different conditions or scenarios require different mathematical models. So, let's embark on this journey to unravel the intricacies of piecewise function evaluation and enhance our mathematical prowess.

Understanding Piecewise Functions

Piecewise functions are defined by different formulas or functions for different intervals of their domain. To understand this better, let’s break down the given piecewise function:

f(x)=\left\{
\begin{array}{l}
\frac{3 x}{2}+8, x < -6 \\
-3 x-2,-4 \leq x \leq 3 \\
4 x+4, x > 3
\end{array}
\right.

This function ${ f(x) }$ has three different rules, each applicable for a specific range of ${ x }$ values.

• For ${ x < -6 }$, the function is defined as ${ f(x) = \frac{3x}{2} + 8 }$.
• For ${ -4 \leq x \leq 3 }$, the function is defined as ${ f(x) = -3x - 2 }$.
• For ${ x > 3 }$, the function is defined as ${ f(x) = 4x + 4 }$.

When evaluating a piecewise function for a particular value of ${ x }$, it's crucial to first identify which interval ${ x }$ falls into. This identification determines which function rule to apply. Understanding these intervals and their corresponding functions is the first step in accurately evaluating piecewise functions. The concept of piecewise functions is widely used in various mathematical and computational applications. For instance, in computer graphics, piecewise functions are used to define curves and shapes by combining different mathematical expressions for different segments of the shape. Similarly, in economics, piecewise functions can model tax brackets, where the tax rate changes based on income levels. Therefore, mastering the understanding and evaluation of piecewise functions is crucial for a wide range of disciplines. By grasping the fundamental principles of these functions, you will be equipped to solve complex problems and gain a deeper appreciation for the versatility of mathematics in modeling real-world phenomena. Let's continue exploring the specific example of ${ f(3) }$ and delve into the practical steps of evaluating this piecewise function.

Determining the Correct Interval

To evaluate ${ f(3) }$, we need to determine which interval contains the value ${ x = 3 }$. Looking at the intervals defined in our piecewise function:

• The first interval is ${ x < -6 }$, which means all ${ x }$ values less than -6. Since 3 is not less than -6, we can rule out this interval.
• The second interval is ${ -4 \leq x \leq 3 }$, which includes all ${ x }$ values between -4 and 3, inclusive. This means 3 is within this interval.
• The third interval is ${ x > 3 }$, which includes all ${ x }$ values greater than 3. Since 3 is not greater than 3, we can rule out this interval.

Therefore, ${ x = 3 }$ falls into the interval ${ -4 \leq x \leq 3 }$. This step is crucial because it dictates which function rule we will use to calculate ${ f(3) }$. Misidentifying the correct interval would lead to an incorrect evaluation of the function. The ability to accurately determine the relevant interval is a fundamental skill when working with piecewise functions. This skill ensures that the appropriate function rule is applied, leading to a correct and meaningful result. Imagine, for instance, applying the wrong tax bracket to your income calculation; the outcome would be significantly different and incorrect. Similarly, in engineering or physics, using the wrong equation for a particular condition can lead to inaccurate predictions or designs. Therefore, the importance of correctly identifying the interval cannot be overstated. It forms the bedrock of accurate piecewise function evaluation and is essential for problem-solving in various scientific and mathematical contexts. Now that we have correctly determined the interval for ${ x = 3 }$, we can proceed to apply the corresponding function rule to calculate the value of ${ f(3) }$.

Applying the Correct Function Rule

Since ${ x = 3 }$ falls into the interval ${ -4 \leq x \leq 3 }$, we use the corresponding function rule:

f(x) = -3x - 2

Now, substitute ${ x = 3 }$ into this function:

f(3) = -3(3) - 2

This substitution is a direct application of the function rule to the specific value of ${ x }$ that we are evaluating. By replacing ${ x }$ with 3 in the equation, we are setting the stage for the final calculation that will yield the value of ${ f(3) }$. The process of substitution is a fundamental step in evaluating any function, and in the case of piecewise functions, it's particularly important to ensure that the substitution is performed within the correct function rule. This meticulous approach guarantees that the result we obtain is accurate and reflects the true behavior of the function at the given point. As we move forward, the arithmetic operations will reveal the numerical value of ${ f(3) }$, providing us with a concrete answer to our evaluation problem. The next step involves simplifying the expression to arrive at the final solution. So, let's proceed with the calculation and uncover the value of ${ f(3) }$.

Calculating the Value

Now, let's perform the arithmetic:

f(3) = -3(3) - 2 = -9 - 2 = -11

So, ${ f(3) = -11 }$. This is the value of the piecewise function at ${ x = 3 }$. The calculation involves simple multiplication and subtraction, but the order of operations is crucial to ensure the correct result. First, we multiply -3 by 3, which gives us -9. Then, we subtract 2 from -9, resulting in -11. This final value represents the output of the piecewise function when the input is 3. It's a single, definitive answer that encapsulates the behavior of the function at this specific point. In the context of piecewise functions, the calculated value is significant because it demonstrates how the function transitions between different rules across its domain. Each value corresponds to a specific segment of the function's graph, creating a composite representation of its overall behavior. The accuracy of this calculation is paramount, as it forms the basis for further analysis and interpretation of the function's properties. Therefore, meticulous attention to detail and a firm grasp of arithmetic principles are essential for successfully evaluating piecewise functions. Now that we have determined the value of ${ f(3) }$, we can confidently conclude our evaluation process.

Final Answer

Therefore, the value of the piecewise function ${ f(x) }$ at ${ x = 3 }$ is ${ -11 }$. This result is obtained by carefully identifying the correct interval for ${ x = 3 }$ and applying the corresponding function rule. The process we followed highlights the importance of understanding the definition of piecewise functions and the steps required to evaluate them accurately. From identifying the appropriate interval to applying the correct function rule and performing the necessary calculations, each step contributes to the final result. This systematic approach can be applied to evaluate any piecewise function at any given point within its domain. Moreover, the concept of piecewise functions extends beyond academic exercises and finds practical applications in various fields. For instance, in computer programming, piecewise functions can be used to define conditional logic, where different actions are taken based on different input values. Similarly, in engineering, piecewise functions can model systems that exhibit different behaviors under different conditions. Therefore, mastering the evaluation of piecewise functions not only enhances mathematical proficiency but also provides a valuable tool for problem-solving in real-world scenarios. With a solid understanding of this concept, you are well-equipped to tackle a wide range of mathematical challenges and applications.