Evaluate $f(-4$\] For The Piecewise Function:$ f(x)=\left\{ \begin{array}{lr} |2x+7|, & X \leq -4 \\ 1+x^2, & -4 \ \textless \ X \leq 1 \\ 6, & 1 \ \textless \ X \ \textless \ 3 \\ \frac{1}{3}x+8, & X \geq

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. These functions are commonly used in mathematics, physics, and engineering to model real-world phenomena. In this article, we will evaluate the piecewise function $f(x)$ at $x = -4$.

The Piecewise Function

The given piecewise function is:

$$f(x)=\left\{ \begin{array}{lr} |2x+7|, & x \leq -4 \\ 1+x^2, & -4 \ \textless \ x \leq 1 \\ 6, & 1 \ \textless \ x \ \textless \ 3 \\ \frac{1}{3}x+8, & x \geq 3 \end{array} \right.$$

Evaluating the Function at $x = -4$

To evaluate the function at $x = -4$, we need to determine which sub-function is applicable. Since $x = -4$ is less than or equal to $-4$, we will use the first sub-function: $|2x+7|$.

Step 1: Substitute $x = -4$ into the Sub-Function

Substituting $x = -4$ into the sub-function $|2x+7|$, we get:

$$|2(-4)+7| = |-8+7| = |-1| = 1$$

Step 2: Evaluate the Absolute Value

The absolute value of $-1$ is $1$. Therefore, the value of the function at $x = -4$ is $1$.

Conclusion

In conclusion, the value of the piecewise function $f(x)$ at $x = -4$ is $1$. This is because the first sub-function $|2x+7|$ is applicable at $x = -4$, and the absolute value of $-1$ is $1$.

Understanding Piecewise Functions

Piecewise functions are a powerful tool in mathematics, allowing us to model complex phenomena with multiple sub-functions. By understanding how to evaluate these functions, we can gain insights into the behavior of real-world systems.

Key Takeaways

• Piecewise functions are defined by multiple sub-functions, each applied to a specific interval of the domain.
• To evaluate a piecewise function, we need to determine which sub-function is applicable based on the value of $x$.
• The absolute value function is used to model real-world phenomena where the magnitude of a quantity is important.

Real-World Applications

Piecewise functions have numerous real-world applications, including:

• Physics: Piecewise functions are used to model the motion of objects, such as the trajectory of a projectile or the vibration of a spring.
• Engineering: Piecewise functions are used to model complex systems, such as the behavior of electrical circuits or the flow of fluids.
• Economics: Piecewise functions are used to model economic systems, such as the behavior of supply and demand curves.

Conclusion

Introduction

In our previous article, we evaluated the piecewise function $f(x)$ at $x = -4$. In this article, we will answer some frequently asked questions about piecewise functions.

Q: What is a piecewise function?

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?

To determine which sub-function to use, you need to evaluate the value of $x$ and determine which interval it falls into. Then, you can use the corresponding sub-function to evaluate the function.

Q: What is the difference between a piecewise function and a regular function?

The main difference between a piecewise function and a regular function is that a piecewise function is defined by multiple sub-functions, while a regular function is defined by a single function.

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

Yes, piecewise functions can be used to model real-world phenomena where the behavior of the system changes at specific points.

Q: How do I graph a piecewise function?

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

Q: Can I use a piecewise function to solve a system of equations?

Yes, piecewise functions can be used to solve a system of equations where the behavior of the system changes at specific points.

Q: What are some common types of piecewise functions?

Some common types of piecewise functions include:

• Step functions: These are piecewise functions that have a constant value for a specific interval.
• Piecewise linear functions: These are piecewise functions that have a linear function for a specific interval.
• Piecewise quadratic functions: These are piecewise functions that have a quadratic function for a specific interval.

Q: How do I evaluate a piecewise function at a specific point?

To evaluate a piecewise function at a specific point, you need to determine which sub-function is applicable and then evaluate the function using that sub-function.

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

Yes, piecewise functions can be used to model a system with multiple variables where the behavior of the system changes at specific points.

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

Some real-world applications of piecewise functions include:

• Physics: Piecewise functions are used to model the motion of objects, such as the trajectory of a projectile or the vibration of a spring.
• Engineering: Piecewise functions are used to model complex systems, such as the behavior of electrical circuits or the flow of fluids.
• Economics: Piecewise functions are used to model economic systems, such as the behavior of supply and demand curves.

Conclusion

In conclusion, piecewise functions are a powerful tool in mathematics, allowing us to model complex phenomena with multiple sub-functions. By understanding how to evaluate and graph these functions, we can gain insights into the behavior of real-world systems and make informed decisions.

Additional Resources

For more information on piecewise functions, including tutorials and examples, please visit the following resources:

• Mathway: A online math problem solver that can help you evaluate and graph piecewise functions.
• Khan Academy: A online learning platform that offers video tutorials and practice exercises on piecewise functions.
• Wolfram Alpha: A online calculator that can help you evaluate and graph piecewise functions.