Study The Function Below And Then Answer The Questions That Follow.${ F(x)=\left{\begin{array}{cl} -x-2, & X\ \textless \ -2 \ -x^2, & -2\ \textless \ X\ \textless \ 0 \ X, & X \geq 0 \end{array}\right. }$1. What Is The Domain Of

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 mathematics, physics, and engineering to model real-world phenomena that exhibit different behaviors in different regions. In this article, we will study a specific piecewise function and answer a series of questions related to its domain, range, and behavior.

The Piecewise Function

The piecewise function we will be studying is defined as:

${ f(x)=\left\{\begin{array}{cl} -x-2, & x\ \textless \ -2 \\ -x^2, & -2\ \textless \ x\ \textless \ 0 \\ x, & x \geq 0 \end{array}\right. }$

This function has three sub-functions, each defined on a specific interval of the domain. The first sub-function, $-x-2$, is defined for $x<-2$. The second sub-function, $-x^2$, is defined for $-2<x<0$. The third sub-function, $x$, is defined for $x\geq 0$.

Domain of the Function

To find the domain of the function, we need to consider the intervals on which each sub-function is defined. The first sub-function is defined for $x<-2$, the second sub-function is defined for $-2<x<0$, and the third sub-function is defined for $x\geq 0$. Therefore, the domain of the function is the union of these intervals:

${ \text{Domain} = (-\infty, -2) \cup (-2, 0) \cup [0, \infty) }$

Range of the Function

To find the range of the function, we need to consider the possible values that each sub-function can take. The first sub-function, $-x-2$, takes values in the interval $(-\infty, -2)$. The second sub-function, $-x^2$, takes values in the interval $[0, \infty)$. The third sub-function, $x$, takes values in the interval $[0, \infty)$. Therefore, the range of the function is the union of these intervals:

${ \text{Range} = (-\infty, -2) \cup [0, \infty) }$

Behavior of the Function

To understand the behavior of the function, we need to consider how each sub-function behaves on its respective interval. The first sub-function, $-x-2$, is a linear function that decreases as $x$ increases. The second sub-function, $-x^2$, is a quadratic function that decreases as $x$ increases. The third sub-function, $x$, is a linear function that increases as $x$ increases.

Questions and Answers

1. What is the domain of the function?

The domain of the function is the union of the intervals $(-\infty, -2)$, $(-2, 0)$, and $[0, \infty)$.

2. What is the range of the function?

The range of the function is the union of the intervals $(-\infty, -2)$ and $[0, \infty)$.

3. How does the function behave on the interval $(-\infty, -2)$?

The function behaves like $-x-2$, which is a linear function that decreases as $x$ increases.

4. How does the function behave on the interval $(-2, 0)$?

The function behaves like $-x^2$, which is a quadratic function that decreases as $x$ increases.

5. How does the function behave on the interval $[0, \infty)$?

The function behaves like $x$, which is a linear function that increases as $x$ increases.

Conclusion

In this article, we studied a piecewise function and answered a series of questions related to its domain, range, and behavior. We found that the domain of the function is the union of the intervals $(-\infty, -2)$, $(-2, 0)$, and $[0, \infty)$. We also found that the range of the function is the union of the intervals $(-\infty, -2)$ and $[0, \infty)$. Finally, we analyzed the behavior of the function on each interval and found that it behaves like a linear or quadratic function.

References

• [1] "Piecewise Functions" by Math Open Reference
• [2] "Piecewise Functions" by Khan Academy
• [3] "Piecewise Functions" by Wolfram MathWorld

Further Reading

• "Piecewise Functions: A Comprehensive Guide" by Mathematics.org
• "Piecewise Functions: Applications and Examples" by Math.com
• "Piecewise Functions: Theory and Practice" by Springer.com
  Piecewise Functions: A Q&A Guide =====================================

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 mathematics, physics, and engineering to model real-world phenomena that exhibit different behaviors in different regions. In this article, we will answer a series of frequently asked questions about piecewise functions.

Q&A

Q1: What is a piecewise function?

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.

Q2: How do I define a piecewise function?

To define a piecewise function, you need to specify the sub-functions and the intervals on which they are defined. For example:

${ f(x)=\left\{\begin{array}{cl} -x-2, & x\ \textless \ -2 \\ -x^2, & -2\ \textless \ x\ \textless \ 0 \\ x, & x \geq 0 \end{array}\right. }$

Q3: What is the domain of a piecewise function?

The domain of a piecewise function is the union of the intervals on which each sub-function is defined.

Q4: How do I find the range of a piecewise function?

To find the range of a piecewise function, you need to consider the possible values that each sub-function can take. For example:

${ \text{Range} = (-\infty, -2) \cup [0, \infty) }$

Q5: How do I graph a piecewise function?

To graph a piecewise function, you need to graph each sub-function on its respective interval. For example:

• Graph the sub-function $-x-2$ on the interval $(-\infty, -2)$.
• Graph the sub-function $-x^2$ on the interval $(-2, 0)$.
• Graph the sub-function $x$ on the interval $[0, \infty)$.

Q6: Can I use piecewise functions to model real-world phenomena?

Yes, piecewise functions can be used to model real-world phenomena that exhibit different behaviors in different regions.

Q7: How do I use piecewise functions in calculus?

Piecewise functions can be used in calculus to find derivatives and integrals. For example:

• Find the derivative of the piecewise function $f(x)=\left\{\begin{array}{cl} -x-2, & x\ \textless \ -2 \\ -x^2, & -2\ \textless \ x\ \textless \ 0 \\ x, & x \geq 0 \end{array}\right.$.
• Find the integral of the piecewise function $f(x)=\left\{\begin{array}{cl} -x-2, & x\ \textless \ -2 \\ -x^2, & -2\ \textless \ x\ \textless \ 0 \\ x, & x \geq 0 \end{array}\right.$.

Q8: Can I use piecewise functions to solve systems of equations?

Yes, piecewise functions can be used to solve systems of equations. For example:

• Solve the system of equations $x+y=2$ and $x-y=1$ using a piecewise function.

Q9: How do I use piecewise functions in computer science?

Piecewise functions can be used in computer science to model complex systems and solve problems. For example:

• Use a piecewise function to model a complex system and find the optimal solution.

Q10: Can I use piecewise functions to model economic systems?

Yes, piecewise functions can be used to model economic systems. For example:

• Use a piecewise function to model a supply and demand curve and find the equilibrium price.

Conclusion

In this article, we answered a series of frequently asked questions about piecewise functions. We covered topics such as defining piecewise functions, finding the domain and range, graphing piecewise functions, and using piecewise functions in calculus, computer science, and economics.

References

• [1] "Piecewise Functions" by Math Open Reference
• [2] "Piecewise Functions" by Khan Academy
• [3] "Piecewise Functions" by Wolfram MathWorld

Further Reading

• "Piecewise Functions: A Comprehensive Guide" by Mathematics.org
• "Piecewise Functions: Applications and Examples" by Math.com
• "Piecewise Functions: Theory and Practice" by Springer.com