Which Graph Represents The Following Piecewise Defined Function?$\[ G(x) = \begin{cases} \frac{1}{2} X + 3, & \text{if } X \ \textless \ -2 \\ 2, & \text{if } -2 \leq X \leq 3 \\ 2x - 3, & \text{if } X \ \textgreater \ 3 \end{cases}

Introduction

In mathematics, a piecewise defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. These functions are often represented graphically, and understanding how to identify the correct graph for a given piecewise defined function is an essential skill for students of mathematics. In this article, we will explore how to determine which graph represents the following piecewise defined function:

${ g(x) = \begin{cases} \frac{1}{2} x + 3, & \text{if } x \ \textless \ -2 \\ 2, & \text{if } -2 \leq x \leq 3 \\ 2x - 3, & \text{if } x \ \textgreater \ 3 \end{cases} }$

Understanding Piecewise Defined Functions

A piecewise defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The function is defined as follows:

${ f(x) = \begin{cases} f_1(x), & \text{if } x \in I_1 \\ f_2(x), & \text{if } x \in I_2 \\ f_3(x), & \text{if } x \in I_3 \end{cases} }$

where $f_1(x)$, $f_2(x)$, and $f_3(x)$ are the sub-functions, and $I_1$, $I_2$, and $I_3$ are the intervals of the domain.

Analyzing the Given Piecewise Defined Function

The given piecewise defined function is:

${ g(x) = \begin{cases} \frac{1}{2} x + 3, & \text{if } x \ \textless \ -2 \\ 2, & \text{if } -2 \leq x \leq 3 \\ 2x - 3, & \text{if } x \ \textgreater \ 3 \end{cases} }$

To determine which graph represents this function, we need to analyze each sub-function and its corresponding interval.

Sub-Function 1: $\frac{1}{2} x + 3$

This sub-function is defined for $x < -2$. We can graph this function by plotting the points $(x, \frac{1}{2} x + 3)$ for $x < -2$. The graph of this function is a line with a slope of $\frac{1}{2}$ and a y-intercept of $3$.

Sub-Function 2: $2$

This sub-function is defined for $-2 \leq x \leq 3$. Since this is a constant function, its graph is a horizontal line at $y = 2$.

Sub-Function 3: $2x - 3$

This sub-function is defined for $x > 3$. We can graph this function by plotting the points $(x, 2x - 3)$ for $x > 3$. The graph of this function is a line with a slope of $2$ and a y-intercept of $-3$.

Graphing the Piecewise Defined Function

To graph the piecewise defined function, we need to combine the graphs of the three sub-functions. The graph of the function is a combination of the three lines:

• The line with a slope of $\frac{1}{2}$ and a y-intercept of $3$ for $x < -2$
• The horizontal line at $y = 2$ for $-2 \leq x \leq 3$
• The line with a slope of $2$ and a y-intercept of $-3$ for $x > 3$

Conclusion

In conclusion, the graph of the piecewise defined function $g(x)$ is a combination of three lines:

• The line with a slope of $\frac{1}{2}$ and a y-intercept of $3$ for $x < -2$
• The horizontal line at $y = 2$ for $-2 \leq x \leq 3$
• The line with a slope of $2$ and a y-intercept of $-3$ for $x > 3$

This graph represents the given piecewise defined function.

Example Questions

1. What is the value of the function $g(x)$ at $x = -4$?
2. What is the value of the function $g(x)$ at $x = 2$?
3. What is the value of the function $g(x)$ at $x = 5$?

Answer Key

1. $g(-4) = \frac{1}{2} (-4) + 3 = -2 + 3 = 1$
2. $g(2) = 2$
3. $g(5) = 2(5) - 3 = 10 - 3 = 7$

Discussion

The graph of a piecewise defined function is a combination of the graphs of the sub-functions. Each sub-function is defined for a specific interval of the domain, and the graph of the function is a combination of the graphs of the sub-functions.

In this article, we analyzed the given piecewise defined function and determined which graph represents the function. We also provided example questions and answers to help students understand the concept of piecewise defined functions.

References

• [1] "Piecewise Defined Functions" by Math Open Reference
• [2] "Graphing Piecewise Defined Functions" by Khan Academy

Additional Resources

• [1] "Piecewise Defined Functions" by Wolfram MathWorld
• [2] "Graphing Piecewise Defined Functions" by Purplemath

Frequently Asked Questions

Q1: What is a piecewise defined function?

A1: A piecewise defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q2: How do I graph a piecewise defined function?

A2: To graph a piecewise defined function, you need to graph each sub-function separately and then combine the graphs. Each sub-function is defined for a specific interval of the domain.

Q3: What is the difference between a piecewise defined function and a continuous function?

A3: A piecewise defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. A continuous function is a function that can be drawn without lifting the pencil from the paper.

Q4: How do I determine which graph represents a piecewise defined function?

A4: To determine which graph represents a piecewise defined function, you need to analyze each sub-function and its corresponding interval. You can graph each sub-function separately and then combine the graphs.

Q5: Can a piecewise defined function be continuous?

A5: Yes, a piecewise defined function can be continuous. However, the function must be continuous at the points where the sub-functions meet.

Q6: How do I find the value of a piecewise defined function at a specific point?

A6: To find the value of a piecewise defined function at a specific point, you need to determine which sub-function is defined at that point and evaluate the sub-function at that point.

Q7: Can a piecewise defined function have multiple sub-functions with the same value?

A7: Yes, a piecewise defined function can have multiple sub-functions with the same value. However, the sub-functions must be defined for different intervals of the domain.

Q8: How do I graph a piecewise defined function with multiple sub-functions?

A8: To graph a piecewise defined function with multiple sub-functions, you need to graph each sub-function separately and then combine the graphs. Each sub-function is defined for a specific interval of the domain.

Q9: Can a piecewise defined function be represented by a single equation?

A9: No, a piecewise defined function cannot be represented by a single equation. The function is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q10: How do I determine the domain of a piecewise defined function?

A10: To determine the domain of a piecewise defined function, you need to analyze each sub-function and its corresponding interval. The domain of the function is the union of the intervals where the sub-functions are defined.

Example Questions

1. What is the value of the function $g(x)$ at $x = -4$?
2. What is the value of the function $g(x)$ at $x = 2$?
3. What is the value of the function $g(x)$ at $x = 5$?

Answer Key

1. $g(-4) = \frac{1}{2} (-4) + 3 = -2 + 3 = 1$
2. $g(2) = 2$
3. $g(5) = 2(5) - 3 = 10 - 3 = 7$

Discussion

The graph of a piecewise defined function is a combination of the graphs of the sub-functions. Each sub-function is defined for a specific interval of the domain, and the graph of the function is a combination of the graphs of the sub-functions.

In this article, we provided answers to frequently asked questions about piecewise defined functions. We also provided example questions and answers to help students understand the concept of piecewise defined functions.

References

• [1] "Piecewise Defined Functions" by Math Open Reference
• [2] "Graphing Piecewise Defined Functions" by Khan Academy

Additional Resources

• [1] "Piecewise Defined Functions" by Wolfram MathWorld
• [2] "Graphing Piecewise Defined Functions" by Purplemath

Note: The references and additional resources provided are for informational purposes only and are not affiliated with this article.