Which Graph Represents The Following Piecewise Defined Function?$[ G(x) = \left{ \begin{array}{cc} x^2, & X \ \textless \ 0 \ \frac{1}{2}x, & 0 \ \textless \ X \leq 4 \ x, & X \ \textgreater \ 4 \end{array} \right.

Mar 8, 2025 by ADMIN 221 views

**Which Graph Represents the Piecewise Defined Function?**

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 sub-functions are often referred to as "pieces" of the function, and they are combined to form the overall function. In this article, we will explore a piecewise defined function and determine which graph represents it.

The Piecewise Defined Function

The piecewise defined function is given by:

${ g(x) = \left\{ \begin{array}{cc} x^2, & x \ \textless \ 0 \\ \frac{1}{2}x, & 0 \ \textless \ x \leq 4 \\ x, & x \ \textgreater \ 4 \end{array} \right. }$

This function has three sub-functions:

For $x < 0$ , the function is $x^2$ .
For $0 < x \leq 4$ , the function is $\frac{1}{2}x$ .
For $x > 4$ , the function is $x$ .

Graphing the Function

To graph the function, we need to graph each sub-function separately and then combine them.

Graphing $x^2$ for $x < 0$

The graph of $x^2$ is a parabola that opens upwards. Since we are only interested in the interval $x < 0$ , we will only graph the left half of the parabola.

### Graphing $\frac{1}{2}x$ for $0 < x \leq 4$
The graph of  $\frac{1}{2}x$  is a line with a slope of  $\frac{1}{2}$ . Since we are only interested in the interval  $0 &lt; x \leq 4$ , we will only graph the line segment from  $x = 0$  to  $x = 4$ .
Graphing  $x$  for  $x &gt; 4$ 
The graph of  $x$  is a line with a slope of 1. Since we are only interested in the interval  $x &gt; 4$ , we will only graph the line segment from  $x = 4$  to the right.
Combining the Graphs
To combine the graphs, we need to find the intersection points between the sub-functions. The intersection points are:

 $x = 0$ , where  $x^2 = \frac{1}{2}x$ 
 $x = 4$ , where  $\frac{1}{2}x = x$ 

The graph of the piecewise defined function is a combination of the three sub-functions, with the intersection points marked.
Conclusion
In conclusion, the graph of the piecewise defined function is a combination of three sub-functions:  $x^2$  for  $x &lt; 0$ ,  $\frac{1}{2}x$  for  $0 &lt; x \leq 4$ , and  $x$  for  $x &gt; 4$ . The graph is a parabola for  $x &lt; 0$ , a line segment for  $0 &lt; x \leq 4$ , and a line segment for  $x &gt; 4$ . The intersection points between the sub-functions are  $x = 0$  and  $x = 4$ .
Key Takeaways

A piecewise defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.
The graph of a piecewise defined function is a combination of the sub-functions, with the intersection points marked.
To graph a piecewise defined function, we need to graph each sub-function separately and then combine them.

Further Reading
For further reading on piecewise defined functions, we recommend the following resources:

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

References

[1] Khan Academy. (n.d.). Piecewise Functions. Retrieved from <https://www.khanacademy.org/math/algebra/x2f4f7c0d-piecewise-functions&gt;
[2] Mathway. (n.d.). Piecewise Functions. Retrieved from <https://www.mathway.com/subjects/piecewise-functions&gt;
[3] Wolfram MathWorld. (n.d.). Piecewise Functions. Retrieved from <https://mathworld.wolfram.com/PiecewiseFunction.html&gt;&lt;br/&gt;
Frequently Asked Questions (FAQs) About Piecewise Defined Functions
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Q: What is a piecewise defined function?
A: A piecewise defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.
Q: How do I graph a piecewise defined function?
A: To graph a piecewise defined function, you need to graph each sub-function separately and then combine them. You also need to find the intersection points between the sub-functions.
Q: What are the intersection points between sub-functions?
A: The intersection points between sub-functions are the points where the sub-functions meet. These points are used to combine the sub-functions into a single graph.
Q: How do I find the intersection points between sub-functions?
A: To find the intersection points between sub-functions, you need to set the sub-functions equal to each other and solve for the variable.
Q: What is the difference between a piecewise defined function and a continuous function?
A: A piecewise defined function is a function that is defined by multiple sub-functions, while a continuous function is a function that has no gaps or jumps in its graph.
Q: Can a piecewise defined function be continuous?
A: Yes, a piecewise defined function can be continuous if the sub-functions meet at the intersection points.
Q: How do I determine if a piecewise defined function is continuous?
A: To determine if a piecewise defined function is continuous, you need to check if the sub-functions meet at the intersection points and if the graph has no gaps or jumps.
Q: What are some common applications of piecewise defined functions?
A: Piecewise defined functions are commonly used in physics, engineering, and economics to model real-world phenomena that have different behaviors in different intervals.
Q: Can I use piecewise defined functions in calculus?
A: Yes, piecewise defined functions can be used in calculus to find derivatives and integrals.
Q: How do I find the derivative of a piecewise defined function?
A: To find the derivative of a piecewise defined function, you need to find the derivative of each sub-function separately and then combine them.
Q: How do I find the integral of a piecewise defined function?
A: To find the integral of a piecewise defined function, you need to find the integral of each sub-function separately and then combine them.
Conclusion
In conclusion, piecewise defined functions are an important concept in mathematics that can be used to model real-world phenomena. By understanding how to graph and analyze piecewise defined functions, you can apply this knowledge to a wide range of fields, including physics, engineering, and economics.
Key Takeaways

A piecewise defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.
To graph a piecewise defined function, you need to graph each sub-function separately and then combine them.
The intersection points between sub-functions are used to combine the sub-functions into a single graph.
Piecewise defined functions can be continuous if the sub-functions meet at the intersection points.
Piecewise defined functions are commonly used in physics, engineering, and economics to model real-world phenomena.

Further Reading
For further reading on piecewise defined functions, we recommend the following resources:

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

References

[1] Khan Academy. (n.d.). Piecewise Functions. Retrieved from <https://www.khanacademy.org/math/algebra/x2f4f7c0d-piecewise-functions&gt;
[2] Mathway. (n.d.). Piecewise Functions. Retrieved from <https://www.mathway.com/subjects/piecewise-functions&gt;
[3] Wolfram MathWorld. (n.d.). Piecewise Functions. Retrieved from <https://mathworld.wolfram.com/PiecewiseFunction.html&gt;

Introduction

The Piecewise Defined Function

Graphing the Function

Graphing x2x^2x2 for x<0x < 0x<0

Graphing xxx for x &gt; 4

Combining the Graphs

Conclusion

Key Takeaways

Further Reading

References

Q: What is a piecewise defined function?

Q: How do I graph a piecewise defined function?

Q: What are the intersection points between sub-functions?

Q: How do I find the intersection points between sub-functions?

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

Q: Can a piecewise defined function be continuous?

Q: How do I determine if a piecewise defined function is continuous?

Q: What are some common applications of piecewise defined functions?

Q: Can I use piecewise defined functions in calculus?

Q: How do I find the derivative of a piecewise defined function?

Q: How do I find the integral of a piecewise defined function?

Conclusion

Key Takeaways

Further Reading

References

Graphing $x^2$ for $x < 0$

Graphing $x$ for $x > 4$