Select The Correct Answer.Consider Function { F $} . . . [ 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

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Introduction

In mathematics, a piecewise 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 overall function. Piecewise 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)={2x,x \textless 0−x2−4x+1,0 \textless x \textless 212x+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. }.

Understanding Piecewise Functions

A piecewise function is defined as:

f(x)={f1(x),x∈D1f2(x),x∈D2f3(x),x∈D3⋮fn(x),x∈Dn{ f(x) = \left\{ \begin{array}{ll} f_1(x), & x \in D_1 \\ f_2(x), & x \in D_2 \\ f_3(x), & x \in D_3 \\ \vdots \\ f_n(x), & x \in D_n \end{array} \right. }

where:

  • f1(x),f2(x),...,fn(x){ f_1(x), f_2(x), ..., f_n(x) } are the sub-functions
  • D1,D2,...,Dn{ D_1, D_2, ..., D_n } are the intervals of the domain where each sub-function is applied

The Given Function

The given function is:

f(x)={2x,x \textless 0−x2−4x+1,0 \textless x \textless 212x+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. }

This function has three sub-functions:

  • f1(x)=2x{ f_1(x) = 2^x } for x \textless 0{ x \ \textless \ 0 }
  • f2(x)=−x2−4x+1{ f_2(x) = -x^2 - 4x + 1 } for 0 \textless x \textless 2{ 0 \ \textless \ x \ \textless \ 2 }
  • f3(x)=12x+3{ f_3(x) = \frac{1}{2}x + 3 } for x \textgreater 2{ x \ \textgreater \ 2 }

Evaluating the Function

To evaluate the function at a given point, we need to determine which sub-function is applicable. We can do this by checking the value of x{ x } and determining which interval it falls into.

For example, if we want to evaluate the function at x=−1{ x = -1 }, we would use the first sub-function, f1(x)=2x{ f_1(x) = 2^x }, since −1 \textless 0{ -1 \ \textless \ 0 }.

Similarly, if we want to evaluate the function at x=1.5{ x = 1.5 }, we would use the second sub-function, f2(x)=−x2−4x+1{ f_2(x) = -x^2 - 4x + 1 }, since 1.5 \textless 2{ 1.5 \ \textless \ 2 }.

Finally, if we want to evaluate the function at x=3{ x = 3 }, we would use the third sub-function, f3(x)=12x+3{ f_3(x) = \frac{1}{2}x + 3 }, since 3 \textgreater 2{ 3 \ \textgreater \ 2 }.

Graphing the Function

To graph the function, we need to graph each sub-function separately and then combine them into a single graph.

The graph of the first sub-function, f1(x)=2x{ f_1(x) = 2^x }, is an exponential curve that increases rapidly as x{ x } decreases.

The graph of the second sub-function, f2(x)=−x2−4x+1{ f_2(x) = -x^2 - 4x + 1 }, is a quadratic curve that opens downward and has a vertex at x=−2{ x = -2 }.

The graph of the third sub-function, f3(x)=12x+3{ f_3(x) = \frac{1}{2}x + 3 }, is a linear curve that increases as x{ x } increases.

Conclusion

In conclusion, piecewise functions are a powerful tool for modeling real-world phenomena that exhibit different behaviors in different regions. The given function, f(x)={2x,x \textless 0−x2−4x+1,0 \textless x \textless 212x+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. }, is a classic example of a piecewise function, with three sub-functions that are applied to different intervals of the domain. By understanding how to evaluate and graph piecewise functions, we can gain a deeper appreciation for the mathematical concepts that underlie many real-world phenomena.

References

  • [1] "Piecewise Functions" by Math Open Reference
  • [2] "Evaluating Piecewise Functions" by Khan Academy
  • [3] "Graphing Piecewise Functions" by Purplemath

Further Reading

  • "Piecewise Functions: A Comprehensive Guide" by Math Is Fun
  • "Evaluating and Graphing Piecewise Functions" by IXL
  • "Piecewise Functions: A Tutorial" by Wolfram Alpha
    Evaluating Piecewise Functions: A Comprehensive Guide ===========================================================

Q&A: Evaluating Piecewise Functions

Q: What is a piecewise function?

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

Q: How do I evaluate a piecewise function?

A: To evaluate a piecewise function, you need to determine which sub-function is applicable by checking the value of x{ x } and determining which interval it falls into.

Q: What if I want to evaluate a piecewise function at a point that is not in any of the intervals?

A: If you want to evaluate a piecewise function at a point that is not in any of the intervals, you need to check if the point is in the domain of the function. If it is not, then the function is not defined at that point.

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 them into a single graph.

Q: What if I want to find the derivative of a piecewise function?

A: To find the derivative of a piecewise function, you need to find the derivative of each sub-function separately and then combine them into a single derivative.

Q: What if I want to find the integral of a piecewise function?

A: To find the integral of a piecewise function, you need to find the integral of each sub-function separately and then combine them into a single integral.

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

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

Q: What are some examples of piecewise functions in real-world applications?

A: Some examples of piecewise functions in real-world applications include:

  • Modeling the cost of a product based on the quantity ordered
  • Modeling the temperature of a substance based on the time of day
  • Modeling the speed of a vehicle based on the distance traveled

Q: How do I determine which sub-function is applicable when evaluating a piecewise function?

A: To determine which sub-function is applicable when evaluating a piecewise function, you need to check the value of x{ x } and determine which interval it falls into.

Q: What if I want to find the inverse of a piecewise function?

A: To find the inverse of a piecewise function, you need to find the inverse of each sub-function separately and then combine them into a single inverse.

Q: Can I use a piecewise function to model a system with multiple states?

A: Yes, piecewise functions can be used to model a system with multiple states.

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 check the value of x{ x } and determine which interval it falls into
  • Failing to graph each sub-function separately and then combine them into a single graph
  • Failing to find the derivative and integral of each sub-function separately and then combine them into a single derivative and integral

Conclusion

In conclusion, piecewise functions are a powerful tool for modeling real-world phenomena that exhibit different behaviors in different regions. By understanding how to evaluate and graph piecewise functions, we can gain a deeper appreciation for the mathematical concepts that underlie many real-world phenomena.

References

  • [1] "Piecewise Functions" by Math Open Reference
  • [2] "Evaluating Piecewise Functions" by Khan Academy
  • [3] "Graphing Piecewise Functions" by Purplemath

Further Reading

  • "Piecewise Functions: A Comprehensive Guide" by Math Is Fun
  • "Evaluating and Graphing Piecewise Functions" by IXL
  • "Piecewise Functions: A Tutorial" by Wolfram Alpha