Read the definition carefully

Oct 13, 2025 by miodeya.com 109 views

Introduction

In mathematics, a piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. These sub-functions are often referred to as "pieces" of the function, and they are used to define the function's behavior over different intervals. In this article, we will explore the concept of piecewise-defined functions and determine the values of a specific function, g(x), at three different points: x = 0, x = 3, and x = 5.

The Piecewise-Defined Function g(x)

The function g(x) is defined as:

g(x) = \left{ \begin{array}{rr} -17, & x \ \textless \ 3 \ 5x, & x \geq 3 \end{array} \right.

This function has two pieces: one for x < 3 and one for x ≥ 3. The first piece is a constant function that returns -17 for all x < 3, while the second piece is a linear function that returns 5x for all x ≥ 3.

Determining the Value of g(0)

To determine the value of g(0), we need to look at the definition of the function. Since 0 < 3, we use the first piece of the function, which is -17. Therefore, g(0) = -17.

Determining the Value of g(3)

To determine the value of g(3), we need to look at the definition of the function. Since 3 ≥ 3, we use the second piece of the function, which is 5x. Therefore, g(3) = 5(3) = 15.

Determining the Value of g(5)

To determine the value of g(5), we need to look at the definition of the function. Since 5 ≥ 3, we use the second piece of the function, which is 5x. Therefore, g(5) = 5(5) = 25.

Conclusion

In conclusion, we have determined the values of the piecewise-defined function g(x) at three different points: x = 0, x = 3, and x = 5. We used the definition of the function to determine the value of each point, and we found that g(0) = -17, g(3) = 15, and g(5) = 25.

Understanding Piecewise-Defined Functions

Piecewise-defined functions are an important concept in mathematics, and they are used to model real-world phenomena. They are particularly useful when dealing with functions that have different behaviors over different intervals. By understanding how to work with piecewise-defined functions, we can better analyze and solve problems in mathematics and other fields.

Real-World Applications of Piecewise-Defined Functions

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

Economics: Piecewise-defined functions can be used to model the behavior of economic systems, such as the relationship between supply and demand.
Physics: Piecewise-defined functions can be used to model the behavior of physical systems, such as the motion of objects under different forces.
Computer Science: Piecewise-defined functions can be used to model the behavior of computer systems, such as the relationship between input and output.

Tips for Working with Piecewise-Defined Functions

When working with piecewise-defined functions, it's essential to:

Read the definition carefully: Make sure you understand the definition of the function and the intervals over which each piece is defined.
Use the correct piece: Make sure you use the correct piece of the function for each interval.
Check your work: Double-check your work to ensure that you have used the correct piece of the function and that your answer is correct.

Common Mistakes to Avoid

When working with piecewise-defined functions, it's essential to avoid the following common mistakes:

Using the wrong piece: Make sure you use the correct piece of the function for each interval.
Not checking your work: Double-check your work to ensure that you have used the correct piece of the function and that your answer is correct.
Not reading the definition carefully: Make sure you understand the definition of the function and the intervals over which each piece is defined.

Conclusion

Q: What is a piecewise-defined function?

A: A piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q: How do I determine which piece of the function to use?

A: To determine which piece of the function to use, you need to look at the definition of the function and the intervals over which each piece is defined. You then use the correct piece of the function for each interval.

Q: What is the difference between a piecewise-defined function and a regular function?

A: The main difference between a piecewise-defined function and a regular function is that a piecewise-defined function has multiple sub-functions, each applied to a specific interval of the domain, while a regular function has a single function that is applied to the entire domain.

Q: Can I use a piecewise-defined function to model real-world phenomena?

A: Yes, piecewise-defined functions can be used to model real-world phenomena. They are particularly useful when dealing with functions that have different behaviors over different intervals.

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

A: To graph a piecewise-defined function, you need to graph each piece of the function separately and then combine them to form the final graph.

Q: Can I use a piecewise-defined function to solve optimization problems?

A: Yes, piecewise-defined functions can be used to solve optimization problems. They are particularly useful when dealing with functions that have different behaviors over different intervals.

Q: How do I determine the domain of a piecewise-defined function?

A: To determine the domain of a piecewise-defined function, you need to look at the definition of the function and the intervals over which each piece is defined. You then combine the intervals to form the final domain.

Q: Can I use a piecewise-defined function to model the behavior of a system over time?

A: Yes, piecewise-defined functions can be used to model the behavior of a system over time. They are particularly useful when dealing with systems that have different behaviors over different time intervals.

Q: How do I determine the range of a piecewise-defined function?

A: To determine the range of a piecewise-defined function, you need to look at the definition of the function and the intervals over which each piece is defined. You then combine the ranges of each piece to form the final range.

Q: Can I use a piecewise-defined function to model the behavior of a system under different conditions?

A: Yes, piecewise-defined functions can be used to model the behavior of a system under different conditions. They are particularly useful when dealing with systems that have different behaviors under different conditions.

Q: How do I determine the derivative of a piecewise-defined function?

A: To determine the derivative of a piecewise-defined function, you need to look at the definition of the function and the intervals over which each piece is defined. You then use the chain rule and the product rule to find the derivative of each piece and combine them to form the final derivative.

Q: Can I use a piecewise-defined function to model the behavior of a system under different forces?

A: Yes, piecewise-defined functions can be used to model the behavior of a system under different forces. They are particularly useful when dealing with systems that have different behaviors under different forces.

Conclusion

In conclusion, piecewise-defined functions are a powerful tool for modeling real-world phenomena and solving optimization problems. By understanding how to work with piecewise-defined functions, you can better analyze and solve problems in mathematics and other fields. Remember to read the definition carefully, use the correct piece, and check your work to avoid common mistakes.