Select The Correct Answer.Consider Function F F F . F ( X ) = { 2 X , X \textless 0 − X 2 − 4 X + 1 , 0 \textless X \textless 2 1 2 X + 3 , X \textgreater 2 F(x)=\left\{\begin{array}{ll} 2^x, & X\ \textless \ 0 \\ -x^2-4x+1, & 0\ \textless \ X\ \textless \ 2 \\ \frac{1}{2}x+3, & X\ \textgreater \ 2 \end{array}\right. F ( X ) = ⎩ ⎨ ⎧ ​ 2 X , − X 2 − 4 X + 1 , 2 1 ​ X + 3 , ​ X \textless 0 0 \textless X \textless 2 X \textgreater 2 ​ Which Statement Is

Introduction

In mathematics, piecewise functions are a type of function that is defined by multiple sub-functions, each applied to a specific interval of the domain. These functions are commonly used to model real-world phenomena that exhibit different behaviors in different regions. In this article, we will explore the concept of piecewise functions, with a focus on the given function $f(x)$.

The Given Function

The function $f(x)$ is defined as:

$$f(x)=\left\{\begin{array}{ll} 2^x, & x\ \textless \ 0 \\ -x^2-4x+1, & 0\ \textless \ x\ \textless \ 2 \\ \frac{1}{2}x+3, & x\ \textgreater \ 2 \end{array}\right.$$

This function is a piecewise function, meaning it is defined by three different sub-functions, each applied to a specific interval of the domain.

Understanding Piecewise Functions

A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The intervals are usually defined by a set of rules or conditions, and the sub-functions are defined separately for each interval.

Key Characteristics of Piecewise Functions

1. Multiple Sub-Functions: Piecewise functions are defined by multiple sub-functions, each applied to a specific interval of the domain.
2. Interval-Based Definition: The intervals are usually defined by a set of rules or conditions, and the sub-functions are defined separately for each interval.
3. Domain and Range: The domain and range of a piecewise function are defined by the intervals and sub-functions.

Evaluating the Given Function

To evaluate the given function $f(x)$, we need to determine which sub-function to use for a given value of $x$. We can do this by checking the value of $x$ against the intervals defined in the function.

Case 1: $x < 0$

If $x < 0$, then we use the sub-function $2^x$. This sub-function is defined for all values of $x$ less than 0.

Case 2: $0 < x < 2$

If $0 < x < 2$, then we use the sub-function $-x^2-4x+1$. This sub-function is defined for all values of $x$ between 0 and 2.

Case 3: $x > 2$

If $x > 2$, then we use the sub-function $\frac{1}{2}x+3$. This sub-function is defined for all values of $x$ greater than 2.

Which Statement is Correct?

The question asks us to determine which statement is correct. To do this, we need to evaluate the given function $f(x)$ and determine which sub-function to use for a given value of $x$.

Conclusion

In conclusion, the given function $f(x)$ is a piecewise function defined by three sub-functions, each applied to a specific interval of the domain. To evaluate the function, we need to determine which sub-function to use for a given value of $x$. We can do this by checking the value of $x$ against the intervals defined in the function.

Final Answer

The final answer is not provided in this article, as it depends on the specific value of $x$ and the corresponding sub-function.

References

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

Additional Resources

• [1] "Piecewise Functions." Wolfram MathWorld, mathworld.wolfram.com/PiecewiseFunction.html.
• [2] "Piecewise Functions." MIT OpenCourseWare, ocw.mit.edu/courses/mathematics/18-01-calculus-i-fall-2006/lecture-notes/lec18.pdf.
  Evaluating Piecewise Functions: A Comprehensive Q&A =====================================================

Introduction

In our previous article, we explored the concept of piecewise functions, with a focus on the given function $f(x)$. In this article, we will provide a comprehensive Q&A section to help you better understand piecewise functions and how to evaluate them.

Q&A Section

Q1: What is a piecewise function?

A1: A piecewise function is a type of function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q2: How do I determine which sub-function to use for a given value of x?

A2: To determine which sub-function to use, you need to check the value of x against the intervals defined in the function. For example, if x < 0, you use the sub-function 2^x.

Q3: What are the key characteristics of piecewise functions?

A3: The key characteristics of piecewise functions are:

• Multiple sub-functions
• Interval-based definition
• Domain and range defined by intervals and sub-functions

Q4: How do I evaluate a piecewise function?

A4: To evaluate a piecewise function, you need to determine which sub-function to use for a given value of x. You can do this by checking the value of x against the intervals defined in the function.

Q5: What is the domain and range of a piecewise function?

A5: The domain and range of a piecewise function are defined by the intervals and sub-functions. For example, if the function is defined for x < 0, 0 < x < 2, and x > 2, the domain is all real numbers, and the range is the set of all possible values of the sub-functions.

Q6: Can I have multiple sub-functions with the same interval?

A6: No, you cannot have multiple sub-functions with the same interval. Each interval must have a unique sub-function.

Q7: How do I graph a piecewise function?

A7: To graph a piecewise function, you need to graph each sub-function separately and then combine them. You can use a graphing calculator or software to help you graph the function.

Q8: Can I have a piecewise function with an infinite number of sub-functions?

A8: Yes, you can have a piecewise function with an infinite number of sub-functions. However, this is not common in mathematics and is usually only used in advanced mathematical contexts.

Q9: How do I find the derivative of a piecewise function?

A9: To find the derivative of a piecewise function, you need to find the derivative of each sub-function separately and then combine them. You can use the chain rule and the product rule to find the derivative of each sub-function.

Q10: Can I have a piecewise function with a sub-function that is not continuous?

A10: No, you cannot have a piecewise function with a sub-function that is not continuous. Each sub-function must be continuous at the endpoints of the interval.

Conclusion

In conclusion, piecewise functions are a powerful tool in mathematics that can be used to model real-world phenomena. By understanding how to evaluate and graph piecewise functions, you can better analyze and solve mathematical problems.

Final Tips

• Make sure to check the value of x against the intervals defined in the function before evaluating the function.
• Use a graphing calculator or software to help you graph the function.
• Be careful when finding the derivative of a piecewise function, as you need to find the derivative of each sub-function separately.

References

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

Additional Resources

• [1] "Piecewise Functions." Wolfram MathWorld, mathworld.wolfram.com/PiecewiseFunction.html.
• [2] "Piecewise Functions." MIT OpenCourseWare, ocw.mit.edu/courses/mathematics/18-01-calculus-i-fall-2006/lecture-notes/lec18.pdf.