Given The Function F ( X ) = { X + 4 For − 3 ≤ X ≤ 0 − 2 X + 7 For 0 \textless X ≤ 6 F(x) = \left\{ \begin{array}{lll} x+4 & \text{for} & -3 \leq X \leq 0 \\ -2x+7 & \text{for} & 0 \ \textless \ X \leq 6 \end{array} \right. F ( X ) = { X + 4 − 2 X + 7 ​ For For ​ − 3 ≤ X ≤ 0 0 \textless X ≤ 6 ​ , Find The Following Values:4. F ( − 2 F(-2 F ( − 2 ] 5. F ( 0 F(0 F ( 0 ] 6.

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. In this article, we will explore the function $f(x) = \left\{ \begin{array}{lll} x+4 & \text{for} & -3 \leq x \leq 0 \\ -2x+7 & \text{for} & 0 \ \textless \ x \leq 6 \end{array} \right.$ and evaluate its values at specific points.

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 function is typically defined as:

$$f(x) = \left\{ \begin{array}{lll} f_1(x) & \text{for} & x \in [a, b] \\ f_2(x) & \text{for} & x \in [c, d] \\ \vdots & & \vdots \end{array} \right.$$

In this case, the function $f(x)$ is defined as:

$$f(x) = \left\{ \begin{array}{lll} x+4 & \text{for} & -3 \leq x \leq 0 \\ -2x+7 & \text{for} & 0 \ \textless \ x \leq 6 \end{array} \right.$$

Evaluating $f(-2)$

To evaluate $f(-2)$, we need to determine which sub-function to use. Since $-2$ is less than $-3$, we will use the first sub-function, $f_1(x) = x+4$.

$$f(-2) = (-2) + 4 = 2$$

Evaluating $f(0)$

To evaluate $f(0)$, we need to determine which sub-function to use. Since $0$ is equal to the lower bound of the second interval, we will use the second sub-function, $f_2(x) = -2x+7$.

$$f(0) = -2(0) + 7 = 7$$

Evaluating $f(6)$

To evaluate $f(6)$, we need to determine which sub-function to use. Since $6$ is less than or equal to $6$, we will use the second sub-function, $f_2(x) = -2x+7$.

$$f(6) = -2(6) + 7 = -5$$

Conclusion

In this article, we evaluated the function $f(x) = \left\{ \begin{array}{lll} x+4 & \text{for} & -3 \leq x \leq 0 \\ -2x+7 & \text{for} & 0 \ \textless \ x \leq 6 \end{array} \right.$ at specific points. We used the definition of piecewise functions to determine which sub-function to use for each evaluation.

Discussion

• What are some common applications of piecewise functions in real-world problems?
• How do you determine which sub-function to use when evaluating a piecewise function?
• Can you think of any other examples of piecewise functions?

Additional Resources

• Khan Academy: Piecewise Functions
• Math Is Fun: Piecewise Functions
• Wolfram MathWorld: Piecewise Functions
  Evaluating Piecewise Functions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the function $f(x) = \left\{ \begin{array}{lll} x+4 & \text{for} & -3 \leq x \leq 0 \\ -2x+7 & \text{for} & 0 \ \textless \ x \leq 6 \end{array} \right.$ and evaluated its values at specific points. In this article, we will answer some common questions about piecewise functions.

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 which sub-function to use when evaluating a piecewise function?

A: To determine which sub-function to use, you need to evaluate the input value of the function and determine which interval it belongs to. Then, you can use the corresponding sub-function to evaluate the function.

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

A: Piecewise functions are commonly used in real-world problems such as:

• Modeling the cost of a product based on the quantity ordered
• Calculating the area of a shape with multiple sides
• Determining the speed of an object based on its distance traveled

Q: Can I use a piecewise function to model a real-world problem with multiple intervals?

A: Yes, you can use a piecewise function to model a real-world problem with multiple intervals. For example, you can use a piecewise function to model the cost of a product based on the quantity ordered, with different costs for different intervals of quantity.

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 to form the piecewise function. You can use a graphing calculator or software to help you graph the function.

Q: Can I use a piecewise function to model a real-world problem with a continuous interval?

A: Yes, you can use a piecewise function to model a real-world problem with a continuous interval. For example, you can use a piecewise function to model the cost of a product based on the quantity ordered, with a continuous cost function for all quantities.

Q: How do I evaluate a piecewise function at a point that is not in the domain?

A: If the point is not in the domain, you cannot evaluate the function at that point. You need to check the domain of the function to see if the point is in the domain.

Q: Can I use a piecewise function to model a real-world problem with multiple variables?

A: Yes, you can use a piecewise function to model a real-world problem with multiple variables. For example, you can use a piecewise function to model the cost of a product based on the quantity ordered and the time of year.

Conclusion

In this article, we answered some common questions about piecewise functions. We hope that this article has helped you to better understand piecewise functions and how to use them to model real-world problems.

Discussion

• What are some other applications of piecewise functions in real-world problems?
• How do you determine which sub-function to use when evaluating a piecewise function?
• Can you think of any other examples of piecewise functions?

Additional Resources

• Khan Academy: Piecewise Functions
• Math Is Fun: Piecewise Functions
• Wolfram MathWorld: Piecewise Functions