Evaluate The Function \[$ F(x) \$\] Given By:$\[ f(x) = \begin{cases} -\frac{3}{4} X + 2, & \text{if } X \ \textless \ 4 \\ \frac{3}{2} X - 1, & \text{if } X \geq 4 \end{cases} \\]a. For \[$ X = 4 \$\], Find \[$ F(4)

Introduction

In mathematics, a piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. These functions are commonly used to model real-world phenomena that exhibit different behaviors in different regions. In this article, we will evaluate the function ${ f(x) }$ given by:

${ f(x) = \begin{cases} -\frac{3}{4} x + 2, & \text{if } x \ \textless \ 4 \\ \frac{3}{2} x - 1, & \text{if } x \geq 4 \end{cases} }$

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:

${ f(x) = \begin{cases} f_1(x), & \text{if } x \in D_1 \\ f_2(x), & \text{if } x \in D_2 \\ \vdots & \text{if } x \in D_n \end{cases} }$

where $f_1(x), f_2(x), \ldots, f_n(x)$ are the sub-functions, and $D_1, D_2, \ldots, D_n$ are the intervals of the domain.

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 greater than or equal to 4, we use the second sub-function:

${ f(4) = \frac{3}{2} (4) - 1 }$

Step-by-Step Solution

1. Identify the sub-function applicable to $x = 4$: Since $x = 4$ is greater than or equal to 4, we use the second sub-function.
2. Substitute $x = 4$ into the second sub-function: ${ f(4) = \frac{3}{2} (4) - 1 }$
3. Simplify the expression: ${ f(4) = 6 - 1 = 5 }$

Conclusion

In this article, we evaluated the function ${ f(x) }$ given by:

${ f(x) = \begin{cases} -\frac{3}{4} x + 2, & \text{if } x \ \textless \ 4 \\ \frac{3}{2} x - 1, & \text{if } x \geq 4 \end{cases} }$

at $x = 4$. We determined that the second sub-function is applicable, and we evaluated the function at $x = 4$ using the second sub-function.

Key Takeaways

• A piecewise function is a function that is 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 $x$.
• We can use the second sub-function to evaluate the function at $x = 4$.

Further Reading

For more information on piecewise functions, we recommend the following resources:

• Khan Academy: Piecewise Functions
• Mathway: Piecewise Functions
• Wolfram MathWorld: Piecewise Functions

Practice Problems

1. Evaluate the function ${ f(x) = \begin{cases} 2x + 1, & \text{if } x \ \textless \ 3 \\ x^2 - 1, & \text{if } x \geq 3 \end{cases} }$ at $x = 3$.
2. Evaluate the function ${ f(x) = \begin{cases} x^2 + 1, & \text{if } x \ \textless \ 2 \\ 2x - 1, & \text{if } x \geq 2 \end{cases} }$ at $x = 2$.

Solutions

1. To evaluate the function at $x = 3$, we need to determine which sub-function is applicable. Since $x = 3$ is less than 3, we use the first sub-function:

${ f(3) = 2(3) + 1 = 7 }$

2. To evaluate the function at $x = 2$, we need to determine which sub-function is applicable. Since $x = 2$ is greater than or equal to 2, we use the second sub-function:

$$

Introduction

In our previous article, we evaluated the function ${ f(x) }$ given by:

${ f(x) = \begin{cases} -\frac{3}{4} x + 2, & \text{if } x \ \textless \ 4 \\ \frac{3}{2} x - 1, & \text{if } x \geq 4 \end{cases} }$

at $x = 4$. In this article, we will provide a comprehensive Q&A guide to help you understand piecewise functions and how to evaluate them.

Q: What is a piecewise function?

A: A piecewise function is a 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 is applicable?

A: To determine which sub-function is applicable, you need to check the value of $x$ and see which interval it falls into. If $x$ is in the first interval, use the first sub-function. If $x$ is in the second interval, use the second sub-function, and so on.

Q: How do I evaluate a piecewise function?

A: To evaluate a piecewise function, you need to follow these steps:

1. Determine which sub-function is applicable based on the value of $x$.
2. Substitute the value of $x$ into the applicable sub-function.
3. Simplify the expression to get the final answer.

Q: What if I have multiple sub-functions with the same interval?

A: If you have multiple sub-functions with the same interval, you need to use the first sub-function that is applicable. For example, if you have:

${ f(x) = \begin{cases} 2x + 1, & \text{if } x \ \textless \ 3 \\ x^2 - 1, & \text{if } x \geq 3 \\ x^3 - 1, & \text{if } x \geq 3 \end{cases} }$

You would use the first sub-function that is applicable, which is $2x + 1$.

Q: Can I have a piecewise function with more than two sub-functions?

A: Yes, you can have a piecewise function with more than two sub-functions. For example:

${ f(x) = \begin{cases} 2x + 1, & \text{if } x \ \textless \ 1 \\ x^2 - 1, & \text{if } 1 \leq x \leq 3 \\ x^3 - 1, & \text{if } x \geq 3 \end{cases} }$

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

Q: Can I have a piecewise function with a single sub-function?

A: Yes, you can have a piecewise function with a single sub-function. For example:

${ f(x) = \begin{cases} 2x + 1, & \text{if } x \ \textless \ 3 \\ 2x + 1, & \text{if } x \geq 3 \end{cases} }$

This is equivalent to the function $f(x) = 2x + 1$.

Conclusion

In this article, we provided a comprehensive Q&A guide to help you understand piecewise functions and how to evaluate them. We covered topics such as determining which sub-function is applicable, evaluating a piecewise function, and graphing a piecewise function.

Key Takeaways

• A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.
• To determine which sub-function is applicable, you need to check the value of $x$ and see which interval it falls into.
• To evaluate a piecewise function, you need to follow the steps of determining which sub-function is applicable, substituting the value of $x$ into the applicable sub-function, and simplifying the expression.

Further Reading

For more information on piecewise functions, we recommend the following resources:

• Khan Academy: Piecewise Functions
• Mathway: Piecewise Functions
• Wolfram MathWorld: Piecewise Functions

Practice Problems

1. Evaluate the function ${ f(x) = \begin{cases} 2x + 1, & \text{if } x \ \textless \ 3 \\ x^2 - 1, & \text{if } x \geq 3 \end{cases} }$ at $x = 3$.
2. Evaluate the function ${ f(x) = \begin{cases} x^2 + 1, & \text{if } x \ \textless \ 2 \\ 2x - 1, & \text{if } x \geq 2 \end{cases} }$ at $x = 2$.

Solutions

1. To evaluate the function at $x = 3$, we need to determine which sub-function is applicable. Since $x = 3$ is greater than or equal to 3, we use the second sub-function:

${ f(3) = x^2 - 1 = 9 - 1 = 8 }$

2. To evaluate the function at $x = 2$, we need to determine which sub-function is applicable. Since $x = 2$ is less than 2, we use the first sub-function:

${ f(2) = x^2 + 1 = 4 + 1 = 5 }$