Evaluate $f(x$\] At $x=-4$ And $x=4$.$\[ f(x) = \begin{cases} x^3 - 4x^2 & \text{if } X \leq -4 \\ 3x^2 & \text{if } -4 \ \textless \ X \ \textless \ 4 \\ x^3 - 45 & \text{if } X \geq 4 \end{cases}

Introduction

Piecewise functions are a type of mathematical function that consists of multiple sub-functions, each defined on a specific interval. 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 $x=-4$ and $x=4$.

The Piecewise Function

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

${ f(x) = \begin{cases} x^3 - 4x^2 & \text{if } x \leq -4 \\ 3x^2 & \text{if } -4 \ \textless \ x \ \textless \ 4 \\ x^3 - 45 & \text{if } x \geq 4 \end{cases} }$

Evaluating $f(x)$ at $x=-4$

To evaluate $f(x)$ at $x=-4$, we need to determine which sub-function is defined on the interval $x \leq -4$. In this case, the sub-function $x^3 - 4x^2$ is defined on this interval.

Substituting $x=-4$ into the sub-function, we get:

${ f(-4) = (-4)^3 - 4(-4)^2 }$

Expanding the expression, we get:

${ f(-4) = -64 - 4(16) }$

Simplifying the expression, we get:

${ f(-4) = -64 - 64 }$

${ f(-4) = -128 }$

Evaluating $f(x)$ at $x=4$

To evaluate $f(x)$ at $x=4$, we need to determine which sub-function is defined on the interval $x \geq 4$. In this case, the sub-function $x^3 - 45$ is defined on this interval.

Substituting $x=4$ into the sub-function, we get:

${ f(4) = (4)^3 - 45 }$

Expanding the expression, we get:

${ f(4) = 64 - 45 }$

Simplifying the expression, we get:

${ f(4) = 19 }$

Conclusion

In this article, we evaluated the piecewise function $f(x)$ at $x=-4$ and $x=4$. We determined which sub-function was defined on each interval and substituted the values of $x$ into the corresponding sub-function. The results were $f(-4)=-128$ and $f(4)=19$.

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 piecewise functions, we can gain insight into the behavior of these functions and make predictions about real-world phenomena.

Tips for Evaluating Piecewise Functions

When evaluating piecewise functions, it's essential to determine which sub-function is defined on each interval. This can be done by examining the conditions that define each sub-function. Once you've determined which sub-function is defined on each interval, you can substitute the values of $x$ into the corresponding sub-function.

Common Mistakes to Avoid

When evaluating piecewise functions, it's easy to make mistakes. Here are a few common mistakes to avoid:

• Not determining which sub-function is defined on each interval: This can lead to incorrect results.
• Not substituting the values of $x$ into the correct sub-function: This can also lead to incorrect results.
• Not simplifying the expression: This can make it difficult to understand the result.

Real-World Applications

Piecewise functions have many real-world applications. Here are a few examples:

• Modeling population growth: Piecewise functions can be used to model population growth, taking into account factors such as birth rates, death rates, and migration.
• Modeling economic systems: Piecewise functions can be used to model economic systems, taking into account factors such as supply and demand, inflation, and interest rates.
• Modeling physical systems: Piecewise functions can be used to model physical systems, taking into account factors such as friction, gravity, and other forces.

Conclusion

Introduction

In our previous article, we evaluated the piecewise function $f(x)$ at $x=-4$ and $x=4$. We also discussed the importance of understanding piecewise functions and provided tips for evaluating them. In this article, we will answer some frequently asked questions about piecewise functions.

Q&A

Q: What is a piecewise function?

A: A piecewise function is a type of mathematical function that consists of multiple sub-functions, each defined on a specific interval.

Q: How do I determine which sub-function is defined on each interval?

A: To determine which sub-function is defined on each interval, you need to examine the conditions that define each sub-function. For example, if the function is defined as:

${ f(x) = \begin{cases} x^3 - 4x^2 & \text{if } x \leq -4 \\ 3x^2 & \text{if } -4 \ \textless \ x \ \textless \ 4 \\ x^3 - 45 & \text{if } x \geq 4 \end{cases} }$

You would examine the conditions $x \leq -4$, $-4 \ \textless \ x \ \textless \ 4$, and $x \geq 4$ to determine which sub-function is defined on each interval.

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 is defined on the interval containing $x$. Once you've determined which sub-function is defined on the interval, you can substitute the value of $x$ into the corresponding sub-function.

Q: What are some common mistakes to avoid when evaluating piecewise functions?

A: Some common mistakes to avoid when evaluating piecewise functions include:

• Not determining which sub-function is defined on each interval
• Not substituting the values of $x$ into the correct sub-function
• Not simplifying the expression

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

A: Piecewise functions have many real-world applications, including:

• Modeling population growth
• Modeling economic systems
• Modeling physical systems

Q: How do I graph a piecewise function?

A: To graph a piecewise function, you need to graph each sub-function on its corresponding interval. You can use a graphing calculator or software to help you graph the function.

Q: Can I use piecewise functions to model complex phenomena?

A: Yes, piecewise functions can be used to model complex phenomena. By breaking down the phenomenon into multiple sub-functions, you can create a more accurate model of the phenomenon.

Q: How do I determine the domain of a piecewise function?

A: To determine the domain of a piecewise function, you need to examine the intervals on which each sub-function is defined. The domain of the function is the set of all values of $x$ for which the function is defined.

Q: Can I use piecewise functions to model periodic phenomena?

A: Yes, piecewise functions can be used to model periodic phenomena. By using sub-functions with periodic behavior, you can create a model of the phenomenon that captures its periodic nature.

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 piecewise functions, we can gain insight into the behavior of these functions and make predictions about real-world phenomena. We hope this Q&A guide has been helpful in answering your questions about piecewise functions.