Calculate Function Values For A Piecewise Function

In mathematics, a piecewise function is a function defined by multiple sub-functions, where each sub-function applies to a certain interval of the main function's domain. These functions are incredibly versatile and appear in various mathematical models and real-world applications. When dealing with piecewise functions, it's crucial to understand how to evaluate the function at different points. The function's definition changes depending on the input value, making it essential to identify the correct sub-function to use for each calculation. In this article, we will delve into the process of calculating function values for a given piecewise function. We will explore the function definition, identify the relevant intervals, and compute the function values for specific input values. This step-by-step approach ensures accuracy and clarity, providing a solid understanding of piecewise function evaluation. Understanding piecewise functions is foundational for more advanced mathematical concepts, including calculus and differential equations. This article aims to provide a clear and comprehensive guide to calculating values for such functions, empowering readers to tackle more complex problems with confidence. Let's explore the intricacies of this particular piecewise function and learn how to calculate its values effectively.

We are given the following piecewise function:

$$f(x)=\left\{\begin{array}{ll} 8 x+9 & x < 0 \\ 8 x+6 & x \geq 0 \end{array}\right.$$

Our task is to calculate the values of the function at three specific points: $f(-1)$, $f(0)$, and $f(2)$. This involves determining which sub-function applies to each input value based on the defined intervals and then performing the corresponding calculation. This exercise highlights the fundamental process of evaluating piecewise functions, demonstrating how different parts of the function definition come into play depending on the input. To successfully solve this problem, we need to carefully examine the conditions under which each sub-function is valid. For $f(-1)$, we need to consider the condition for $x < 0$, while for $f(0)$ and $f(2)$, we look at the condition for $x \geq 0$. By correctly identifying the applicable sub-function for each input, we can accurately compute the function values. This problem provides a practical application of the concept of piecewise functions, reinforcing the understanding of how these functions operate. In the following sections, we will break down each calculation step by step, ensuring a clear and thorough solution.

To calculate $f(-1)$, we first need to determine which part of the piecewise function applies. Since $-1 < 0$, we use the first sub-function, which is $f(x) = 8x + 9$. Substituting $x = -1$ into this equation gives us:

$$f(-1) = 8(-1) + 9$$

Now, we perform the arithmetic:

$$f(-1) = -8 + 9$$

$$f(-1) = 1$$

Thus, the value of the function at $x = -1$ is 1. This calculation demonstrates the direct application of the piecewise function's definition for negative values of $x$. By identifying the correct sub-function and substituting the given value, we arrive at the solution. The process is straightforward, but it underscores the importance of carefully considering the conditions that define each part of the piecewise function. This step-by-step approach ensures accuracy and provides a clear understanding of how piecewise functions behave. In the next section, we will evaluate the function at $x = 0$, where the transition between the two sub-functions occurs. This will further illustrate the nuances of piecewise function evaluation and highlight the significance of the defined intervals.

Next, we need to calculate $f(0)$. According to the piecewise function definition, the second sub-function, $f(x) = 8x + 6$, applies when $x \geq 0$. Since $0 \geq 0$, we use this sub-function. Substituting $x = 0$ into the equation, we get:

$$f(0) = 8(0) + 6$$

Now, we perform the arithmetic:

$$f(0) = 0 + 6$$

$$f(0) = 6$$

Therefore, the value of the function at $x = 0$ is 6. This calculation highlights the importance of correctly interpreting the inequality conditions in the piecewise function definition. The condition $x \geq 0$ includes $x = 0$, so we use the second sub-function. This step reinforces the understanding of how the function's value changes at the boundary between the defined intervals. Evaluating the function at the transition point is crucial for understanding the function's behavior and continuity. In this case, we see a distinct value for $f(0)$ based on the second sub-function. In the following section, we will evaluate the function at $x = 2$, which will further demonstrate the application of the second sub-function for positive values of $x$. This comprehensive approach ensures a thorough understanding of how to evaluate piecewise functions at various points.

Now, let's calculate $f(2)$. Since $2 \geq 0$, we again use the second sub-function, $f(x) = 8x + 6$. Substituting $x = 2$ into this equation, we get:

$$f(2) = 8(2) + 6$$

Performing the arithmetic:

$$f(2) = 16 + 6$$

$$f(2) = 22$$

Thus, the value of the function at $x = 2$ is 22. This calculation further demonstrates the use of the second sub-function for positive values of $x$. By consistently applying the correct part of the piecewise function definition, we can accurately determine the function's value at any given point. This step reinforces the understanding of how piecewise functions behave over different intervals of their domain. The calculation of $f(2)$ completes our evaluation of the function at the specified points. We have now calculated $f(-1)$, $f(0)$, and $f(2)$, demonstrating the process of evaluating a piecewise function across different intervals. In the conclusion, we will summarize our findings and highlight the key steps in evaluating piecewise functions.

We have successfully calculated the values of the piecewise function $f(x)=\left{\begin{array}{ll} 8 x+9 & x < 0 \ 8 x+6 & x \geq 0 \end{array}\right.$ at the points $x = -1$, $x = 0$, and $x = 2$. Our calculations yielded the following results:

• $f(-1) = 1$
• $f(0) = 6$
• $f(2) = 22$

These results demonstrate the piecewise nature of the function, with different sub-functions applying to different intervals of the domain. The process involved identifying the correct sub-function based on the input value and then performing the corresponding calculation. This step-by-step approach ensures accuracy and clarity in evaluating piecewise functions. Understanding how to evaluate these functions is crucial for various mathematical applications, including calculus, differential equations, and modeling real-world phenomena. By carefully considering the conditions under which each sub-function applies, we can confidently determine the function's value at any point. This exercise provides a solid foundation for working with piecewise functions and tackling more complex problems involving these versatile mathematical constructs. In the conclusion, we will reiterate the key steps for evaluating piecewise functions and emphasize the importance of this skill in mathematics.

In this article, we have explored the process of calculating function values for a piecewise function. We began with a given piecewise function defined by two sub-functions, each applicable over different intervals. We then calculated the function values at three specific points: $f(-1)$, $f(0)$, and $f(2)$. The key to successfully evaluating a piecewise function lies in correctly identifying the sub-function that applies to the given input value. This involves carefully considering the conditions defined for each sub-function and determining which condition the input value satisfies. Once the correct sub-function is identified, the calculation is straightforward, involving substitution and arithmetic. We found that $f(-1) = 1$, $f(0) = 6$, and $f(2) = 22$. These results illustrate how the function's behavior changes across different intervals, highlighting the piecewise nature of the function. Understanding how to evaluate piecewise functions is essential for various mathematical applications. These functions are used to model situations where the relationship between variables changes depending on the context. From simple step functions to more complex definitions, piecewise functions provide a flexible tool for representing diverse phenomena. By mastering the techniques presented in this article, readers can confidently tackle problems involving piecewise functions and gain a deeper understanding of their role in mathematics. This skill is crucial for further studies in calculus, differential equations, and other advanced mathematical topics. The ability to accurately evaluate these functions is a valuable asset in both academic and practical settings.