Which Graph Represents The Following Piecewise Defined Function?${ g(x) = \begin{cases} x^2, & X \ \textless \ 0 \ \frac{1}{2} X, & 0 \ \textless \ X \leq 4 \ x, & X \ \textgreater \ 4 \end{cases} }$

Mar 12, 2025 by ADMIN 207 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 used to describe the behavior of the function over different intervals. 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:

${ g(x) = \begin{cases} x^2, & x \ \textless \ 0 \\ \frac{1}{2} x, & 0 \ \textless \ x \leq 4 \\ x, & x \ \textgreater \ 4 \end{cases} \}$

This function has three sub-functions, each defined on a specific interval of the domain. The first sub-function, $x^2$ , is defined for $x < 0$ . The second sub-function, $\frac{1}{2} x$ , is defined for $0 < x \leq 4$ . The third sub-function, $x$ , is defined for $x > 4$ .

Graphing the Function

To graph the function, we need to graph each sub-function on its respective interval. Let's start with the first sub-function, $x^2$ . This is a quadratic function that opens upwards, and its graph is a parabola that is symmetric about the y-axis. Since the domain of this sub-function is $x < 0$ , the graph will only be defined for negative values of $x$ .

### Graph of x^2 for x < 0

The graph of x^2 for x < 0 is a parabola that opens upwards and is symmetric about the y-axis. The graph is only defined for negative values of x.

Next, let's graph the second sub-function, $\frac{1}{2} x$ . This is a linear function that has a slope of $\frac{1}{2}$ and a y-intercept of 0. Since the domain of this sub-function is $0 < x \leq 4$ , the graph will only be defined for positive values of $x$ between 0 and 4.

### Graph of 1/2 x for 0 < x <= 4
The graph of 1/2 x for 0 < x <= 4 is a line with a slope of 1/2 and a y-intercept of 0. The graph is only defined for positive values of x between 0 and 4.

Finally, let's graph the third sub-function, $x$ . This is a linear function that has a slope of 1 and a y-intercept of 0. Since the domain of this sub-function is $x > 4$ , the graph will only be defined for positive values of $x$ greater than 4.

### Graph of x for x > 4

The graph of x for x > 4 is a line with a slope of 1 and a y-intercept of 0. The graph is only defined for positive values of x greater than 4.

Combining the Graphs

Now that we have graphed each sub-function on its respective interval, we can combine the graphs to get the final graph of the piecewise defined function. The graph will consist of three parts: the parabola for $x < 0$ , the line for $0 < x \leq 4$ , and the line for $x > 4$ .

### Final Graph of the Piecewise Defined Function
The final graph of the piecewise defined function consists of three parts: the parabola for x < 0, the line for 0 < x <= 4, and the line for x > 4.

Conclusion

In this article, we explored a piecewise defined function and determined which graph represents it. We graphed each sub-function on its respective interval and combined the graphs to get the final graph of the function. The graph consists of three parts: a parabola for $x < 0$ , a line for $0 < x \leq 4$ , and a line for $x > 4$ . This type of function is commonly used in mathematics to describe the behavior of a function over different intervals.

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 consists of multiple parts, each corresponding to a sub-function.
The domain of each sub-function must be specified in order to graph the function correctly.

References

[1] "Calculus: Early Transcendentals" by James Stewart
[2] "Algebra and Trigonometry" by Michael Sullivan
[3] "Mathematics for the Nonmathematician" by Morris Kline
Piecewise Defined Functions: Q&A =====================================

Introduction

In our previous article, we explored the concept of piecewise defined functions and determined which graph represents a given piecewise defined function. In this article, we will answer some frequently asked questions about piecewise defined functions.

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: Why are piecewise defined functions used?

A: Piecewise defined functions are used to describe the behavior of a function over different intervals. They are commonly used in mathematics to model real-world phenomena that have different behaviors in different regions.

Q: How do I graph a piecewise defined function?

A: To graph a piecewise defined function, you need to graph each sub-function on its respective interval. You can use the following steps:

Identify the sub-functions and their respective intervals.
Graph each sub-function on its interval.
Combine the graphs to get the final graph of the function.

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

A: Some common types of piecewise defined functions include:

Step functions: These are piecewise defined functions that have a constant value on each interval.
Piecewise linear functions: These are piecewise defined functions that have a linear function on each interval.
Piecewise quadratic functions: These are piecewise defined functions that have a quadratic function on each interval.

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

A: To determine the domain of a piecewise defined function, you need to identify the intervals on which each sub-function is defined. The domain of the function is the union of these intervals.

Q: Can I use piecewise defined functions to model real-world phenomena?

A: Yes, piecewise defined functions can be used to model real-world phenomena that have different behaviors in different regions. For example, you can use a piecewise defined function to model the temperature of a city over different seasons.

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

A: To evaluate a piecewise defined function at a given point, you need to determine which sub-function is defined at that point. You can use the following steps:

Identify the sub-functions and their respective intervals.
Determine which sub-function is defined at the given point.
Evaluate the sub-function at the given point.

Q: Can I use piecewise defined functions to solve optimization problems?

A: Yes, piecewise defined functions can be used to solve optimization problems. For example, you can use a piecewise defined function to model the cost of producing a product and then use optimization techniques to minimize the cost.

Q: How do I use piecewise defined functions in calculus?

A: Piecewise defined functions are used extensively in calculus to model real-world phenomena and to solve optimization problems. You can use piecewise defined functions to model the behavior of a function over different intervals and to solve optimization problems.

Conclusion

In this article, we answered some frequently asked questions about piecewise defined functions. We hope that this article has provided you with a better understanding of piecewise defined functions and how they can be used to model real-world phenomena.

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.
Piecewise defined functions are used to describe the behavior of a function over different intervals.
You can use piecewise defined functions to model real-world phenomena and to solve optimization problems.

References

[1] "Calculus: Early Transcendentals" by James Stewart
[2] "Algebra and Trigonometry" by Michael Sullivan
[3] "Mathematics for the Nonmathematician" by Morris Kline

Which Graph Represents The Following Piecewise Defined Function?${ g(x) = \begin{cases} x^2, & X \ \textless \ 0 \ \frac{1}{2} X, & 0 \ \textless \ X \leq 4 \ x, & X \ \textgreater \ 4 \end{cases} }$

Introduction

The Piecewise Defined Function

Graphing the Function

Combining the Graphs

Conclusion

Key Takeaways

Further Reading

References

Introduction

Q: What is a piecewise defined function?

Q: Why are piecewise defined functions used?

Q: How do I graph a piecewise defined function?

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

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

Q: Can I use piecewise defined functions to model real-world phenomena?

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

Q: Can I use piecewise defined functions to solve optimization problems?

Q: How do I use piecewise defined functions in calculus?

Conclusion

Key Takeaways

Further Reading

References