Jenny Is Tracking The Monthly Sales Totals For Her Boutique. The Given Piecewise Function Represents The Boutique's Monthly Sales, In Dollars, Where $x$ Represents The Number Of Months Since Jenny Began Tracking The Data.$\[

Introduction

Jenny is the owner of a boutique, and she is tracking the monthly sales totals. The given piecewise function represents the boutique's monthly sales, in dollars, where $x$ represents the number of months since Jenny began tracking the data. In this article, we will analyze the piecewise function and understand how it represents the monthly sales of the boutique.

Understanding Piecewise Functions

A piecewise function is a function that is defined by multiple sub-functions, each of which is applied to a specific interval of the domain. In other words, a piecewise function is a function that is composed of multiple functions, each of which is applied to a specific part of the domain.

The piecewise function that represents the monthly sales of the boutique is given by:

$$\begin{cases} 1000x & \text{if } 0 \leq x \leq 3 \\ 2000 + 500(x-3) & \text{if } 3 < x \leq 6 \\ 3000 + 200(x-6) & \text{if } 6 < x \leq 9 \\ 4000 + 100(x-9) & \text{if } 9 < x \leq 12 \\ 5000 & \text{if } x > 12 \end{cases}$$

Analyzing the Piecewise Function

Let's analyze the piecewise function and understand how it represents the monthly sales of the boutique.

• For the first three months, the sales are $1000x$, which means that the sales are increasing at a rate of $1000 per month.
• For the next three months, the sales are $2000 + 500(x-3)$, which means that the sales are increasing at a rate of $500 per month, but with a base of $2000.
• For the next three months, the sales are $3000 + 200(x-6)$, which means that the sales are increasing at a rate of $200 per month, but with a base of $3000.
• For the next three months, the sales are $4000 + 100(x-9)$, which means that the sales are increasing at a rate of $100 per month, but with a base of $4000.
• For the months after the 12th month, the sales are $5000, which means that the sales are constant at $5000 per month.

Graphing the Piecewise Function

To visualize the piecewise function, we can graph it using a graphing tool or a programming language. The graph of the piecewise function will show the different intervals of the domain and the corresponding values of the function.

Calculating Sales for Specific Months

To calculate the sales for specific months, we can plug in the value of $x$ into the piecewise function. For example, to calculate the sales for the 5th month, we can plug in $x=5$ into the piecewise function:

$$\begin{cases} 1000x & \text{if } 0 \leq x \leq 3 \\ 2000 + 500(x-3) & \text{if } 3 < x \leq 6 \\ 3000 + 200(x-6) & \text{if } 6 < x \leq 9 \\ 4000 + 100(x-9) & \text{if } 9 < x \leq 12 \\ 5000 & \text{if } x > 12 \end{cases}$$

Plugging in $x=5$ into the piecewise function, we get:

$$2000 + 500(5-3) = 2000 + 500(2) = 2000 + 1000 = 3000$$

Therefore, the sales for the 5th month are $3000.

Conclusion

In this article, we analyzed the piecewise function that represents the monthly sales of the boutique. We understood how the piecewise function is defined and how it represents the monthly sales of the boutique. We also graphed the piecewise function and calculated the sales for specific months. The piecewise function provides a useful tool for analyzing the monthly sales of the boutique and making informed decisions about the business.

Future Work

In the future, we can use the piecewise function to analyze the monthly sales of the boutique over a longer period of time. We can also use the piecewise function to make predictions about the future sales of the boutique based on historical data.

References

• [1] "Piecewise Functions." Math Open Reference, mathopenref.com/piecewise.html.
• [2] "Graphing Piecewise Functions." Khan Academy, khanacademy.org/math/algebra/advanced-algebra/graphing-piecewise-functions.

Glossary

• Piecewise function: A function that is defined by multiple sub-functions, each of which is applied to a specific interval of the domain.
• Domain: The set of all possible input values for a function.
• Range: The set of all possible output values for a function.
• Graph: A visual representation of a function.
• Interval: A subset of the domain of a function.

FAQs

• Q: What is a piecewise function?
• A: A piecewise function is a function that is defined by multiple sub-functions, each of which is applied to a specific interval of the domain.
• Q: How do I graph a piecewise function?
• A: To graph a piecewise function, you can use a graphing tool or a programming language.
• Q: How do I calculate the sales for a specific month?
• A: To calculate the sales for a specific month, you can plug in the value of $x$ into the piecewise function.
  Frequently Asked Questions (FAQs) about Piecewise Functions ================================================================

Q: What is a piecewise function?

A: A piecewise function is a function that is defined by multiple sub-functions, each of which is applied to a specific interval of the domain. In other words, a piecewise function is a function that is composed of multiple functions, each of which is applied to a specific part of the domain.

Q: How do I define a piecewise function?

A: To define a piecewise function, you need to specify the sub-functions and the intervals of the domain for each sub-function. For example, the piecewise function:

$$\begin{cases} 1000x & \text{if } 0 \leq x \leq 3 \\ 2000 + 500(x-3) & \text{if } 3 < x \leq 6 \\ 3000 + 200(x-6) & \text{if } 6 < x \leq 9 \\ 4000 + 100(x-9) & \text{if } 9 < x \leq 12 \\ 5000 & \text{if } x > 12 \end{cases}$$

is defined by five sub-functions, each of which is applied to a specific interval of the domain.

Q: How do I graph a piecewise function?

A: To graph a piecewise function, you can use a graphing tool or a programming language. The graph of a piecewise function will show the different intervals of the domain and the corresponding values of the function.

Q: How do I calculate the sales for a specific month?

A: To calculate the sales for a specific month, you can plug in the value of $x$ into the piecewise function. For example, to calculate the sales for the 5th month, you can plug in $x=5$ into the piecewise function:

$$\begin{cases} 1000x & \text{if } 0 \leq x \leq 3 \\ 2000 + 500(x-3) & \text{if } 3 < x \leq 6 \\ 3000 + 200(x-6) & \text{if } 6 < x \leq 9 \\ 4000 + 100(x-9) & \text{if } 9 < x \leq 12 \\ 5000 & \text{if } x > 12 \end{cases}$$

Plugging in $x=5$ into the piecewise function, we get:

$$2000 + 500(5-3) = 2000 + 500(2) = 2000 + 1000 = 3000$$

Therefore, the sales for the 5th month are $3000.

Q: What are some common applications of piecewise functions?

A: Piecewise functions have many applications in mathematics, science, and engineering. Some common applications of piecewise functions include:

• Modeling real-world phenomena, such as population growth or temperature changes
• Analyzing data, such as sales or stock prices
• Solving optimization problems, such as finding the maximum or minimum of a function
• Modeling complex systems, such as electrical circuits or mechanical systems

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

A: To determine the domain of a piecewise function, you need to identify the intervals of the domain for each sub-function. For example, the piecewise function:

$$\begin{cases} 1000x & \text{if } 0 \leq x \leq 3 \\ 2000 + 500(x-3) & \text{if } 3 < x \leq 6 \\ 3000 + 200(x-6) & \text{if } 6 < x \leq 9 \\ 4000 + 100(x-9) & \text{if } 9 < x \leq 12 \\ 5000 & \text{if } x > 12 \end{cases}$$

has a domain of $[0, 12]$.

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

A: To determine the range of a piecewise function, you need to identify the range of each sub-function. For example, the piecewise function:

$$\begin{cases} 1000x & \text{if } 0 \leq x \leq 3 \\ 2000 + 500(x-3) & \text{if } 3 < x \leq 6 \\ 3000 + 200(x-6) & \text{if } 6 < x \leq 9 \\ 4000 + 100(x-9) & \text{if } 9 < x \leq 12 \\ 5000 & \text{if } x > 12 \end{cases}$$

has a range of $[0, 5000]$.

Q: How do I simplify a piecewise function?

A: To simplify a piecewise function, you can combine the sub-functions into a single function. For example, the piecewise function:

$$\begin{cases} 1000x & \text{if } 0 \leq x \leq 3 \\ 2000 + 500(x-3) & \text{if } 3 < x \leq 6 \\ 3000 + 200(x-6) & \text{if } 6 < x \leq 9 \\ 4000 + 100(x-9) & \text{if } 9 < x \leq 12 \\ 5000 & \text{if } x > 12 \end{cases}$$

can be simplified to:

$$f(x) = \begin{cases} 1000x & \text{if } 0 \leq x \leq 3 \\ 2000 + 500(x-3) & \text{if } 3 < x \leq 6 \\ 3000 + 200(x-6) & \text{if } 6 < x \leq 9 \\ 4000 + 100(x-9) & \text{if } 9 < x \leq 12 \\ 5000 & \text{if } x > 12 \end{cases}$$

is equivalent to:

$$f(x) = \begin{cases} 1000x & \text{if } 0 \leq x \leq 3 \\ 2000 + 500x - 1500 & \text{if } 3 < x \leq 6 \\ 3000 + 200x - 1200 & \text{if } 6 < x \leq 9 \\ 4000 + 100x - 900 & \text{if } 9 < x \leq 12 \\ 5000 & \text{if } x > 12 \end{cases}$$

which can be further simplified to:

$$f(x) = \begin{cases} 1000x & \text{if } 0 \leq x \leq 3 \\ 500x - 500 & \text{if } 3 < x \leq 6 \\ 200x - 1000 & \text{if } 6 < x \leq 9 \\ 100x - 2000 & \text{if } 9 < x \leq 12 \\ 5000 & \text{if } x > 12 \end{cases}$$

Q: How do I use piecewise functions in real-world applications?

A: Piecewise functions have many applications in real-world scenarios. Some examples include:

• Modeling population growth or decline
• Analyzing sales or stock prices
• Solving optimization problems, such as finding the maximum or minimum of a function
• Modeling complex systems, such as electrical circuits or mechanical systems

Q: What are some common mistakes to avoid when working with piecewise functions?

A: Some common mistakes to avoid when working with piecewise functions include:

• Failing to specify the intervals of the domain for each sub-function
• Failing to identify the range of each sub-function
• Failing to simplify the piecewise function when possible
• Failing to use the correct notation for piecewise functions

Q: How do I troubleshoot common issues with piecewise functions?

A: Some common issues with piecewise functions include:

• Failing to specify the intervals of the domain for each sub-function
• Failing to identify the range of each sub-function
• Failing to simplify the piecewise function when possible
• Failing to use the correct notation for piecewise functions

To troubleshoot these issues, you can:

• Review the definition of the piecewise function to ensure that the intervals of the domain are specified correctly
• Review the range of each sub-function to ensure that it is correct
• Simplify the piecewise function when possible to make it easier to work with
• Use the correct notation for piecewise functions to avoid confusion

Q: What are some advanced topics related to piecewise functions?

A: Some advanced topics related to piecewise functions include:

• Piecewise functions with multiple sub-functions
• Piecewise functions with complex intervals of the domain
• Piecewise functions with non-linear sub-functions
• Piecewise functions with multiple variables

These topics