Find F ( 5 F(5 F ( 5 ] In This Piecewise Function.${ \begin{aligned} x \leq 6, \quad Y &= 2x - 3 \ x \ \textgreater \ 6, \quad Y &= -\frac{2}{3}x + 13 \end{aligned} }$f(5) = , ?$

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 how to solve piecewise functions, with a focus on finding the value of $f(5)$ in the given piecewise function.

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 using the following notation:

$$f(x) = \begin{cases} f_1(x) & \text{if } x \in [a, b] \\ f_2(x) & \text{if } x \in [c, d] \\ \vdots & \end{cases}$$

In this notation, $f_1(x)$ and $f_2(x)$ are the sub-functions, and $[a, b]$ and $[c, d]$ are the intervals of the domain where each sub-function is applied.

The Given Piecewise Function

The given piecewise function is:

$$\begin{aligned} x \leq 6, \quad y &= 2x - 3 \\ x \ \textgreater \ 6, \quad y &= -\frac{2}{3}x + 13 \end{aligned}$$

This function is defined by two sub-functions: $f_1(x) = 2x - 3$ for $x \leq 6$, and $f_2(x) = -\frac{2}{3}x + 13$ for $x > 6$.

Finding $f(5)$

To find $f(5)$, we need to determine which sub-function is applied to the value $x = 5$. Since $5 \leq 6$, we will use the sub-function $f_1(x) = 2x - 3$.

Step 1: Evaluate the Sub-Function

To evaluate the sub-function $f_1(x) = 2x - 3$ at $x = 5$, we substitute $x = 5$ into the equation:

$$f_1(5) = 2(5) - 3$$

Step 2: Simplify the Expression

To simplify the expression, we multiply $2$ and $5$ to get $10$, and then subtract $3$:

$$f_1(5) = 10 - 3 = 7$$

Conclusion

Therefore, the value of $f(5)$ is $7$.

Why is this Important?

Solving piecewise functions is an important skill in mathematics, as it allows us to analyze and understand complex functions that are defined by multiple sub-functions. By following the steps outlined in this article, you can learn how to solve piecewise functions and apply this knowledge to a wide range of mathematical problems.

Real-World Applications

Piecewise functions have many real-world applications, including:

• Computer Science: Piecewise functions are used in computer science to model complex systems and algorithms.
• Engineering: Piecewise functions are used in engineering to model and analyze complex systems, such as electrical circuits and mechanical systems.
• Economics: Piecewise functions are used in economics to model and analyze complex economic systems, such as supply and demand curves.

Conclusion

In conclusion, solving piecewise functions is an important skill in mathematics that has many real-world applications. By following the steps outlined in this article, you can learn how to solve piecewise functions and apply this knowledge to a wide range of mathematical problems.

Additional Resources

For additional resources on solving piecewise functions, including video tutorials and practice problems, please visit the following websites:

• Khan Academy: Khan Academy offers a comprehensive course on piecewise functions, including video tutorials and practice problems.
• Mathway: Mathway is an online math problem solver that can help you solve piecewise functions and other mathematical problems.
• Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve piecewise functions and other mathematical problems.

Final Thoughts

Introduction

In our previous article, we explored how to solve piecewise functions, with a focus on finding the value of $f(5)$ in the given piecewise function. In this article, we will answer some of the most frequently asked questions about solving 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 know which sub-function to use?

A: To determine which sub-function to use, you need to evaluate the value of $x$ and determine which interval it falls into. For example, if the piecewise function is defined as:

$$\begin{aligned} x \leq 6, \quad y &= 2x - 3 \\ x \ \textgreater \ 6, \quad y &= -\frac{2}{3}x + 13 \end{aligned}$$

If $x = 5$, you would use the sub-function $f_1(x) = 2x - 3$ because $5 \leq 6$.

Q: What if the value of $x$ falls into multiple intervals?

A: If the value of $x$ falls into multiple intervals, you need to use the sub-function that is defined for the interval that contains the value of $x$. For example, if the piecewise function is defined as:

$$\begin{aligned} x \leq 3, \quad y &= 2x - 3 \\ 3 \leq x \leq 6, \quad y &= -\frac{2}{3}x + 13 \\ x \ \textgreater \ 6, \quad y &= x^2 - 4 \end{aligned}$$

If $x = 5$, you would use the sub-function $f_2(x) = -\frac{2}{3}x + 13$ because $5$ falls into the interval $[3, 6]$.

Q: How do I evaluate a piecewise function at a specific value of $x$?

A: To evaluate a piecewise function at a specific value of $x$, you need to determine which sub-function to use and then evaluate the sub-function at the value of $x$. For example, if the piecewise function is defined as:

$$\begin{aligned} x \leq 6, \quad y &= 2x - 3 \\ x \ \textgreater \ 6, \quad y &= -\frac{2}{3}x + 13 \end{aligned}$$

If $x = 5$, you would use the sub-function $f_1(x) = 2x - 3$ and evaluate it at $x = 5$:

$$f_1(5) = 2(5) - 3 = 7$$

Q: What if the piecewise function is defined with multiple sub-functions?

A: If the piecewise function is defined with multiple sub-functions, you need to use the sub-function that is defined for the interval that contains the value of $x$. For example, if the piecewise function is defined as:

$$\begin{aligned} x \leq 3, \quad y &= 2x - 3 \\ 3 \leq x \leq 6, \quad y &= -\frac{2}{3}x + 13 \\ x \ \textgreater \ 6, \quad y &= x^2 - 4 \end{aligned}$$

If $x = 5$, you would use the sub-function $f_2(x) = -\frac{2}{3}x + 13$ because $5$ falls into the interval $[3, 6]$.

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

A: Yes, piecewise functions can be used to model real-world problems. For example, a piecewise function can be used to model the cost of a product based on the quantity ordered. If the quantity ordered is less than or equal to 10, the cost is $2x - 3$. If the quantity ordered is greater than 10, the cost is $-\frac{2}{3}x + 13$.

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 the graphs. For example, if the piecewise function is defined as:

$$\begin{aligned} x \leq 6, \quad y &= 2x - 3 \\ x \ \textgreater \ 6, \quad y &= -\frac{2}{3}x + 13 \end{aligned}$$

You would graph the sub-function $f_1(x) = 2x - 3$ for $x \leq 6$ and the sub-function $f_2(x) = -\frac{2}{3}x + 13$ for $x > 6$.

Conclusion

In conclusion, solving piecewise functions is an important skill in mathematics that has many real-world applications. By following the steps outlined in this article, you can learn how to solve piecewise functions and apply this knowledge to a wide range of mathematical problems.

Additional Resources

For additional resources on solving piecewise functions, including video tutorials and practice problems, please visit the following websites:

• Khan Academy: Khan Academy offers a comprehensive course on piecewise functions, including video tutorials and practice problems.
• Mathway: Mathway is an online math problem solver that can help you solve piecewise functions and other mathematical problems.
• Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve piecewise functions and other mathematical problems.

Final Thoughts

Solving piecewise functions is a challenging but rewarding topic in mathematics. By following the steps outlined in this article, you can learn how to solve piecewise functions and apply this knowledge to a wide range of mathematical problems. Remember to practice regularly and seek help when needed to become proficient in solving piecewise functions.