20. Given The Piecewise Function:$\[ F(x)=\begin{cases} 2x + 7 & \text{if } X \ \textless \ -4 \\ -\frac{1}{4}x & \text{if } X \geq -4 \end{cases} \\]Determine The Following:Domain: $\qquad$Range: $\qquad$

Introduction

Piecewise functions are a type of mathematical function that is defined by multiple sub-functions, each applied to a specific interval of the domain. These functions are commonly used in various mathematical and real-world applications, such as modeling physical systems, analyzing data, and solving optimization problems. In this article, we will focus on analyzing the domain and range of a given piecewise function.

The Piecewise Function

The given piecewise function is defined as:

${ f(x)=\begin{cases} 2x + 7 & \text{if } x \ \textless \ -4 \\ -\frac{1}{4}x & \text{if } x \geq -4 \end{cases} }$

This function has two sub-functions: one for the interval $x < -4$ and another for the interval $x \geq -4$. The first sub-function is a linear function of the form $2x + 7$, while the second sub-function is a linear function of the form $-\frac{1}{4}x$.

Domain Analysis

To determine the domain of the piecewise function, we need to consider the intervals for which each sub-function is defined. The first sub-function is defined for $x < -4$, which means that the domain of this sub-function is the set of all real numbers less than -4. The second sub-function is defined for $x \geq -4$, which means that the domain of this sub-function is the set of all real numbers greater than or equal to -4.

However, since the piecewise function is defined as a single function, we need to consider the intersection of the domains of the two sub-functions. In this case, the intersection of the domains is the set of all real numbers, since both sub-functions are defined for all real numbers.

Therefore, the domain of the piecewise function is the set of all real numbers, denoted as $\mathbb{R}$.

Range Analysis

To determine the range of the piecewise function, we need to consider the possible values of the function for each sub-function. For the first sub-function, $2x + 7$, the possible values are all real numbers greater than or equal to $2(-4) + 7 = -1$. For the second sub-function, $-\frac{1}{4}x$, the possible values are all real numbers less than or equal to $-\frac{1}{4}(-4) = 1$.

However, since the piecewise function is defined as a single function, we need to consider the intersection of the ranges of the two sub-functions. In this case, the intersection of the ranges is the set of all real numbers between -1 and 1, inclusive.

Therefore, the range of the piecewise function is the set of all real numbers between -1 and 1, inclusive, denoted as $[-1, 1]$.

Conclusion

In conclusion, the domain of the piecewise function is the set of all real numbers, denoted as $\mathbb{R}$, and the range of the piecewise function is the set of all real numbers between -1 and 1, inclusive, denoted as $[-1, 1]$.

Discussion

The analysis of the domain and range of a piecewise function is an important aspect of understanding the behavior of the function. By considering the intervals for which each sub-function is defined, we can determine the domain and range of the piecewise function.

In this case, the piecewise function has a domain of $\mathbb{R}$ and a range of $[-1, 1]$. This means that the function is defined for all real numbers and takes on all values between -1 and 1, inclusive.

Example Use Cases

The piecewise function can be used in various real-world applications, such as:

• Modeling the behavior of a physical system, such as a spring-mass system, where the function represents the displacement of the mass over time.
• Analyzing data, such as the temperature of a city over a period of time, where the function represents the temperature as a function of time.
• Solving optimization problems, such as finding the maximum or minimum value of a function subject to certain constraints.

Conclusion

In conclusion, the piecewise function is a powerful tool for modeling and analyzing complex systems. By understanding the domain and range of the function, we can gain insights into the behavior of the system and make informed decisions.

References

• [1] "Piecewise Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/piecewise.html
• [2] "Domain and Range of a Function" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-domain-range/x2f-domain-range/v/domain-and-range-of-a-function

Glossary

• Piecewise function: A function that is defined by multiple sub-functions, each applied to a specific interval of the domain.
• Domain: The set of all input values for which a function is defined.
• Range: The set of all output values that a function can take on.
• Sub-function: A function that is defined for a specific interval of the domain.
• Interval: A set of real numbers that includes all numbers between two given numbers, inclusive.
  Piecewise Function Q&A ==========================

Q: What is a piecewise function?

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

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

A: To determine the domain of a piecewise function, you need to consider the intervals for which each sub-function is defined. The domain of the piecewise function is the intersection of the domains of the sub-functions.

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

A: To determine the range of a piecewise function, you need to consider the possible values of the function for each sub-function. The range of the piecewise function is the intersection of the ranges of the sub-functions.

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

A: A regular function is a function that is defined for all real numbers, whereas a piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q: Can a piecewise function have multiple sub-functions with the same domain?

A: Yes, a piecewise function can have multiple sub-functions with the same domain. In this case, the sub-functions are said to be "overlapping" or "nested".

Q: How do I graph a piecewise function?

A: To graph a piecewise function, you need to graph each sub-function separately and then combine the graphs to form the graph of the piecewise function.

Q: Can a piecewise function be used to model real-world phenomena?

A: Yes, a piecewise function can be used to model real-world phenomena, such as the behavior of a physical system, the temperature of a city over time, or the growth of a population.

Q: How do I use a piecewise function to solve optimization problems?

A: To use a piecewise function to solve optimization problems, you need to define the function and then use calculus techniques, such as finding the derivative and setting it equal to zero, to find the maximum or minimum value of the function.

Q: Can a piecewise function be used to model non-linear relationships?

A: Yes, a piecewise function can be used to model non-linear relationships, such as the relationship between the price of a product and the quantity demanded.

Q: How do I determine the number of sub-functions needed for a piecewise function?

A: To determine the number of sub-functions needed for a piecewise function, you need to consider the complexity of the function and the number of intervals for which the function is defined.

Q: Can a piecewise function be used to model periodic phenomena?

A: Yes, a piecewise function can be used to model periodic phenomena, such as the behavior of a pendulum or the growth of a population over time.

Q: How do I use a piecewise function to model a system with multiple variables?

A: To use a piecewise function to model a system with multiple variables, you need to define the function and then use calculus techniques, such as finding the partial derivatives and setting them equal to zero, to find the maximum or minimum value of the function.

Q: Can a piecewise function be used to model systems with non-linear relationships?

A: Yes, a piecewise function can be used to model systems with non-linear relationships, such as the relationship between the price of a product and the quantity demanded.

Q: How do I determine the number of sub-functions needed for a piecewise function with multiple variables?

A: To determine the number of sub-functions needed for a piecewise function with multiple variables, you need to consider the complexity of the function and the number of intervals for which the function is defined.

Q: Can a piecewise function be used to model systems with periodic phenomena?

A: Yes, a piecewise function can be used to model systems with periodic phenomena, such as the behavior of a pendulum or the growth of a population over time.

Q: How do I use a piecewise function to model a system with multiple sub-functions?

A: To use a piecewise function to model a system with multiple sub-functions, you need to define the function and then use calculus techniques, such as finding the partial derivatives and setting them equal to zero, to find the maximum or minimum value of the function.

Q: Can a piecewise function be used to model systems with non-linear relationships and periodic phenomena?

A: Yes, a piecewise function can be used to model systems with non-linear relationships and periodic phenomena, such as the relationship between the price of a product and the quantity demanded over time.

Conclusion

In conclusion, piecewise functions are a powerful tool for modeling and analyzing complex systems. By understanding the domain and range of the function, we can gain insights into the behavior of the system and make informed decisions.