Evaluate The Function For The Indicated Values Of X X X .${ f(x) = \begin{cases} 2x + 1, & X \leq -5 \ x^2, & -5 \ \textless \ X \ \textless \ 5 \ 3 - X, & X \geq 5 \end{cases} }$1. $f(-10) = \square$2. $f(2) =

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Introduction

In this article, we will evaluate the function f(x)f(x) for the indicated values of xx. The function f(x)f(x) is defined as a piecewise function, which means it has different definitions for different intervals of xx. We will use the given function definition to find the values of f(x)f(x) for the specified values of xx.

Function Definition

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

f(x)={2x+1,x≤−5x2,−5<x<53−x,x≥5{ f(x) = \begin{cases} 2x + 1, & x \leq -5 \\ x^2, & -5 < x < 5 \\ 3 - x, & x \geq 5 \end{cases} }

Evaluating f(−10)f(-10)

To evaluate f(−10)f(-10), we need to determine which part of the function definition applies to x=−10x = -10. Since x=−10x = -10 is less than or equal to −5-5, we will use the first part of the function definition: f(x)=2x+1f(x) = 2x + 1.

Substituting x=−10x = -10 into the function definition, we get:

f(−10)=2(−10)+1f(-10) = 2(-10) + 1

f(−10)=−20+1f(-10) = -20 + 1

f(−10)=−19f(-10) = -19

Therefore, f(−10)=−19f(-10) = -19.

Evaluating f(2)f(2)

To evaluate f(2)f(2), we need to determine which part of the function definition applies to x=2x = 2. Since x=2x = 2 is greater than −5-5 and less than 55, we will use the second part of the function definition: f(x)=x2f(x) = x^2.

Substituting x=2x = 2 into the function definition, we get:

f(2)=(2)2f(2) = (2)^2

f(2)=4f(2) = 4

Therefore, f(2)=4f(2) = 4.

Evaluating f(6)f(6)

To evaluate f(6)f(6), we need to determine which part of the function definition applies to x=6x = 6. Since x=6x = 6 is greater than or equal to 55, we will use the third part of the function definition: f(x)=3−xf(x) = 3 - x.

Substituting x=6x = 6 into the function definition, we get:

f(6)=3−6f(6) = 3 - 6

f(6)=−3f(6) = -3

Therefore, f(6)=−3f(6) = -3.

Conclusion

In this article, we evaluated the function f(x)f(x) for the indicated values of xx. We used the given function definition to find the values of f(x)f(x) for x=−10x = -10, x=2x = 2, and x=6x = 6. The results were f(−10)=−19f(-10) = -19, f(2)=4f(2) = 4, and f(6)=−3f(6) = -3.

Discussion

The function f(x)f(x) is a piecewise function, which means it has different definitions for different intervals of xx. This type of function is commonly used to model real-world situations where the behavior of a system changes at certain points. In this case, the function f(x)f(x) changes its behavior at x=−5x = -5 and x=5x = 5.

The function f(x)f(x) can be used to model a variety of real-world situations, such as the behavior of a physical system over time. For example, the function f(x)f(x) could be used to model the position of an object over time, where the object's behavior changes at certain points.

Applications

The function f(x)f(x) has a variety of applications in mathematics and science. For example, it can be used to model the behavior of a physical system over time, as mentioned earlier. It can also be used to solve optimization problems, where the goal is to maximize or minimize a function over a given interval.

Future Work

In the future, it would be interesting to explore the properties of the function f(x)f(x) in more detail. For example, one could investigate the function's continuity and differentiability at the points where it changes its behavior. Additionally, one could explore the function's behavior as xx approaches infinity or negative infinity.

References

Keywords

  • Piecewise function
  • Function definition
  • Mathematical modeling
  • Optimization problems
  • Continuity and differentiability

Conclusion

In conclusion, the function f(x)f(x) is a piecewise function that has different definitions for different intervals of xx. We evaluated the function for the indicated values of xx and found the values of f(x)f(x) for x=−10x = -10, x=2x = 2, and x=6x = 6. The results were f(−10)=−19f(-10) = -19, f(2)=4f(2) = 4, and f(6)=−3f(6) = -3. The function f(x)f(x) has a variety of applications in mathematics and science, and it can be used to model real-world situations where the behavior of a system changes at certain points.

Introduction

In our previous article, we evaluated the function f(x)f(x) for the indicated values of xx. We used the given function definition to find the values of f(x)f(x) for x=−10x = -10, x=2x = 2, and x=6x = 6. In this article, we will answer some frequently asked questions about the function f(x)f(x) and its evaluation.

Q&A

Q1: What is the function f(x)f(x)?

A1: The function f(x)f(x) is a piecewise function defined as:

f(x)={2x+1,x≤−5x2,−5<x<53−x,x≥5{ f(x) = \begin{cases} 2x + 1, & x \leq -5 \\ x^2, & -5 < x < 5 \\ 3 - x, & x \geq 5 \end{cases} }

Q2: How do I evaluate the function f(x)f(x) for a given value of xx?

A2: To evaluate the function f(x)f(x) for a given value of xx, you need to determine which part of the function definition applies to xx. If xx is less than or equal to −5-5, use the first part of the function definition: f(x)=2x+1f(x) = 2x + 1. If xx is greater than −5-5 and less than 55, use the second part of the function definition: f(x)=x2f(x) = x^2. If xx is greater than or equal to 55, use the third part of the function definition: f(x)=3−xf(x) = 3 - x.

Q3: What is the value of f(−10)f(-10)?

A3: The value of f(−10)f(-10) is −19-19. To find this value, we used the first part of the function definition: f(x)=2x+1f(x) = 2x + 1. Substituting x=−10x = -10 into the function definition, we get:

f(−10)=2(−10)+1f(-10) = 2(-10) + 1

f(−10)=−20+1f(-10) = -20 + 1

f(−10)=−19f(-10) = -19

Q4: What is the value of f(2)f(2)?

A4: The value of f(2)f(2) is 44. To find this value, we used the second part of the function definition: f(x)=x2f(x) = x^2. Substituting x=2x = 2 into the function definition, we get:

f(2)=(2)2f(2) = (2)^2

f(2)=4f(2) = 4

Q5: What is the value of f(6)f(6)?

A5: The value of f(6)f(6) is −3-3. To find this value, we used the third part of the function definition: f(x)=3−xf(x) = 3 - x. Substituting x=6x = 6 into the function definition, we get:

f(6)=3−6f(6) = 3 - 6

f(6)=−3f(6) = -3

Q6: Can I use the function f(x)f(x) to model real-world situations?

A6: Yes, the function f(x)f(x) can be used to model real-world situations where the behavior of a system changes at certain points. For example, the function f(x)f(x) could be used to model the position of an object over time, where the object's behavior changes at certain points.

Q7: What are some applications of the function f(x)f(x)?

A7: The function f(x)f(x) has a variety of applications in mathematics and science. For example, it can be used to model the behavior of a physical system over time, solve optimization problems, and investigate the properties of the function in more detail.

Conclusion

In this article, we answered some frequently asked questions about the function f(x)f(x) and its evaluation. We provided examples of how to evaluate the function for given values of xx and discussed some of the applications of the function in mathematics and science. We hope this article has been helpful in understanding the function f(x)f(x) and its evaluation.

Discussion

The function f(x)f(x) is a piecewise function that has different definitions for different intervals of xx. It can be used to model real-world situations where the behavior of a system changes at certain points. The function has a variety of applications in mathematics and science, and it can be used to solve optimization problems and investigate the properties of the function in more detail.

References

Keywords

  • Piecewise function
  • Function definition
  • Mathematical modeling
  • Optimization problems
  • Continuity and differentiability

Future Work

In the future, it would be interesting to explore the properties of the function f(x)f(x) in more detail. For example, one could investigate the function's continuity and differentiability at the points where it changes its behavior. Additionally, one could explore the function's behavior as xx approaches infinity or negative infinity.

Conclusion

In conclusion, the function f(x)f(x) is a piecewise function that has different definitions for different intervals of xx. We evaluated the function for the indicated values of xx and answered some frequently asked questions about the function and its evaluation. The function has a variety of applications in mathematics and science, and it can be used to model real-world situations where the behavior of a system changes at certain points.