Evaluate The Piecewise-defined Function At The Indicated Values.${ f(x) = \begin{cases} 5 & \text{if } X \leq 6 \ 6x - 4 & \text{if } X \ \textgreater \ 6 \end{cases} }$- { F(-7) = \square $} − \[ - \[ − \[ F(0) = \square

Introduction

Piecewise-defined functions are a fundamental concept in mathematics, particularly in calculus and algebra. These functions are defined by multiple sub-functions, each applicable to a specific interval or domain. In this article, we will delve into the world of piecewise-defined functions, focusing on evaluating them at indicated values. We will explore the concept of piecewise-defined functions, learn how to evaluate them, and provide step-by-step examples to reinforce our understanding.

What are Piecewise-Defined Functions?

A piecewise-defined function is a function that is defined by multiple sub-functions, each applicable to a specific interval or domain. These sub-functions are often referred to as "pieces" of the function. The function is defined as a combination of these pieces, with each piece being a separate function that is applied to a specific interval.

Notation and Representation

Piecewise-defined functions are often represented using a notation that indicates the different sub-functions and their corresponding intervals. This notation is typically written as:

${ f(x) = \begin{cases} f_1(x) & \text{if } x \in I_1 \\ f_2(x) & \text{if } x \in I_2 \\ & \vdots \\ f_n(x) & \text{if } x \in I_n \end{cases} }$

where $f_i(x)$ is the $i$-th sub-function, $I_i$ is the interval corresponding to the $i$-th sub-function, and $n$ is the number of sub-functions.

Evaluating Piecewise-Defined Functions

Evaluating a piecewise-defined function at a specific value involves determining which sub-function is applicable to that value. This is done by checking which interval the value falls into. Once the applicable sub-function is identified, the function is evaluated using that sub-function.

Step-by-Step Example

Let's consider the piecewise-defined function:

${ f(x) = \begin{cases} 5 & \text{if } x \leq 6 \\ 6x - 4 & \text{if } x \ \textgreater \ 6 \end{cases} }$

To evaluate this function at $x = -7$, we need to determine which sub-function is applicable. Since $-7$ is less than $6$, the first sub-function $f_1(x) = 5$ is applicable. Therefore, we evaluate the function as follows:

${ f(-7) = 5 }$

Evaluating f(0)

To evaluate the function at $x = 0$, we need to determine which sub-function is applicable. Since $0$ is less than $6$, the first sub-function $f_1(x) = 5$ is applicable. Therefore, we evaluate the function as follows:

${ f(0) = 5 }$

Evaluating f(7)

To evaluate the function at $x = 7$, we need to determine which sub-function is applicable. Since $7$ is greater than $6$, the second sub-function $f_2(x) = 6x - 4$ is applicable. Therefore, we evaluate the function as follows:

${ f(7) = 6(7) - 4 = 38 }$

Conclusion

Evaluating piecewise-defined functions requires a clear understanding of the function's notation and representation. By identifying the applicable sub-function and evaluating the function using that sub-function, we can determine the value of the function at a specific point. In this article, we have explored the concept of piecewise-defined functions, learned how to evaluate them, and provided step-by-step examples to reinforce our understanding.

Common Mistakes to Avoid

When evaluating piecewise-defined functions, it's essential to avoid common mistakes such as:

• Failing to identify the applicable sub-function
• Evaluating the function using the wrong sub-function
• Not checking the intervals corresponding to each sub-function

By being aware of these common mistakes, we can ensure that our evaluations are accurate and reliable.

Real-World Applications

Piecewise-defined functions have numerous real-world applications, including:

• Modeling physical systems with different behaviors in different intervals
• Representing data with different patterns in different intervals
• Solving problems in engineering, economics, and other fields

By understanding piecewise-defined functions and how to evaluate them, we can apply this knowledge to real-world problems and make informed decisions.

Final Thoughts

Q&A: Evaluating Piecewise-Defined Functions

Q: What is a piecewise-defined function?

A: A piecewise-defined function is a function that is defined by multiple sub-functions, each applicable to a specific interval or domain. These sub-functions are often referred to as "pieces" of the function.

Q: How do I represent a piecewise-defined function?

A: A piecewise-defined function is typically represented using a notation that indicates the different sub-functions and their corresponding intervals. This notation is written as:

${ f(x) = \begin{cases} f_1(x) & \text{if } x \in I_1 \\ f_2(x) & \text{if } x \in I_2 \\ & \vdots \\ f_n(x) & \text{if } x \in I_n \end{cases} }$

Q: How do I evaluate a piecewise-defined function at a specific value?

A: To evaluate a piecewise-defined function at a specific value, you need to determine which sub-function is applicable to that value. This is done by checking which interval the value falls into. Once the applicable sub-function is identified, the function is evaluated using that sub-function.

Q: What if the value I'm evaluating the function at falls into multiple intervals?

A: If the value you're evaluating the function at falls into multiple intervals, you need to use the sub-function that is applicable to the interval that the value falls into first. For example, if the function is defined as:

${ f(x) = \begin{cases} 5 & \text{if } x \leq 6 \\ 6x - 4 & \text{if } x \ \textgreater \ 6 \end{cases} }$

and you want to evaluate the function at $x = 6$, you would use the first sub-function $f_1(x) = 5$.

Q: Can I have multiple sub-functions with the same interval?

A: Yes, you can have multiple sub-functions with the same interval. For example:

${ f(x) = \begin{cases} 5 & \text{if } x \leq 6 \\ 6x - 4 & \text{if } x \ \textgreater \ 6 \\ 3x + 2 & \text{if } x \ \textgreater \ 6 \end{cases} }$

In this case, the second and third sub-functions both apply to the interval $x \ \textgreater \ 6$. You would need to use the sub-function that is applicable to the interval that the value falls into first.

Q: How do I know which sub-function to use?

A: To determine which sub-function to use, you need to check which interval the value falls into. You can do this by comparing the value to the endpoints of the intervals.

Q: Can I have a piecewise-defined function with an infinite number of sub-functions?

A: Yes, you can have a piecewise-defined function with an infinite number of sub-functions. For example:

${ f(x) = \begin{cases} 5 & \text{if } x \leq 6 \\ 6x - 4 & \text{if } 6 \ \textless \ x \leq 7 \\ 3x + 2 & \text{if } 7 \ \textless \ x \leq 8 \\ & \vdots \end{cases} }$

In this case, the function is defined by an infinite number of sub-functions, each applicable to a specific interval.

Q: How do I graph a piecewise-defined function?

A: To graph a piecewise-defined function, you need to graph each sub-function separately and then combine the graphs. You can use a graphing calculator or software to help you graph the function.

Q: Can I have a piecewise-defined function with a single sub-function?

A: Yes, you can have a piecewise-defined function with a single sub-function. For example:

${ f(x) = \begin{cases} 5 & \text{if } x \leq 6 \end{cases} }$

In this case, the function is defined by a single sub-function that applies to the entire domain.

Conclusion

Evaluating piecewise-defined functions requires a clear understanding of the function's notation and representation. By identifying the applicable sub-function and evaluating the function using that sub-function, we can determine the value of the function at a specific point. In this article, we have explored the concept of piecewise-defined functions, learned how to evaluate them, and provided step-by-step examples to reinforce our understanding.