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

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, 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 Piecewise Defined Function

To graph the piecewise defined function, we need to graph each sub-function on its respective interval. We will start by graphing the first sub-function, $x^2$, on the interval $x < 0$.

Graphing $x^2$ on $x < 0$

The graph of $x^2$ on the interval $x < 0$ is a parabola that opens upwards. The vertex of the parabola is at the origin, $(0, 0)$. The parabola is symmetric about the y-axis, and it has a minimum value of $0$ at $x = 0$.

**Graph of $x^2$ on $x < 0$**
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*   The graph is a parabola that opens upwards.
*   The vertex is at the origin, $(0, 0)$.
*   The parabola is symmetric about the y-axis.
*   The minimum value is $0$ at $x = 0$.

Graphing $\frac{1}{2} x$ on $0 < x \leq 4$

The graph of $\frac{1}{2} x$ on the interval $0 < x \leq 4$ is a line with a slope of $\frac{1}{2}$. The line passes through the origin, $(0, 0)$, and has a maximum value of $4$ at $x = 4$.

**Graph of $\frac{1}{2} x$ on $0 < x \leq 4$**
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*   The graph is a line with a slope of $\frac{1}{2}$.
*   The line passes through the origin, $(0, 0)$.
*   The maximum value is $4$ at $x = 4$.

Graphing $x$ on $x > 4$

The graph of $x$ on the interval $x > 4$ is a line with a slope of $1$. The line passes through the point $(4, 4)$ and has a minimum value of $4$ at $x = 4$.

**Graph of $x$ on $x > 4$**
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*   The graph is a line with a slope of $1$.
*   The line passes through the point $(4, 4)$.
*   The minimum value is $4$ at $x = 4$.

Combining the Graphs

To graph the piecewise defined function, we need to combine the graphs of each sub-function on its respective interval. We will start by combining the graphs of $x^2$ and $\frac{1}{2} x$ on the interval $x < 4$.

**Graph of $g(x)$ on $x < 4$**
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*   The graph is a parabola that opens upwards for $x < 0$.
*   The graph is a line with a slope of $\frac{1}{2}$ for $0 < x \leq 4$.

Next, we will combine the graphs of $\frac{1}{2} x$ and $x$ on the interval $x > 4$.

**Graph of $g(x)$ on $x > 4$**
---------------------------

*   The graph is a line with a slope of $\frac{1}{2}$ for $0 < x \leq 4$.
*   The graph is a line with a slope of $1$ for $x > 4$.

Conclusion

In conclusion, the graph of the piecewise defined function $g(x)$ is a combination of 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 $x > 4$. The graph of $g(x)$ is a parabola that opens upwards for $x < 0$, a line with a slope of $\frac{1}{2}$ for $0 < x \leq 4$, and a line with a slope of $1$ for $x > 4$.

Final Answer

The final answer is:

Graph of $g(x)$

• The graph is a parabola that opens upwards for $x < 0$.
• The graph is a line with a slope of $\frac{1}{2}$ for $0 < x \leq 4$.
• The graph is a line with a slope of $1$ for $x > 4$.

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Introduction

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

Q: What is a piecewise defined function?

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.

Q: How do I graph a piecewise defined function?

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

1. Identify the sub-functions and their respective intervals.
2. Graph each sub-function on its interval.
3. Combine the graphs of each sub-function to form the overall graph.

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

Some common types of piecewise defined functions include:

• Step functions: These functions have a constant value on each interval.
• Piecewise linear functions: These functions are linear on each interval.
• Piecewise quadratic functions: These functions are quadratic on each interval.

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

To find 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: How do I find the range of a piecewise defined function?

To find the range of a piecewise defined function, you need to identify the values that each sub-function takes on its interval. The range of the function is the union of these values.

Q: Can I use piecewise defined functions in real-world applications?

Yes, piecewise defined functions can be used in a variety of real-world applications, including:

• Physics: Piecewise defined functions can be used to model the motion of objects under different conditions.
• Engineering: Piecewise defined functions can be used to model the behavior of complex systems.
• Economics: Piecewise defined functions can be used to model the behavior of economic systems.

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

To evaluate a piecewise defined function at a given point, you need to identify the sub-function that is defined on the interval containing the point. You can then evaluate the sub-function at the point.

Q: Can I use piecewise defined functions in calculus?

Yes, piecewise defined functions can be used in calculus. In fact, piecewise defined functions are often used to model complex systems in calculus.

Conclusion

In conclusion, piecewise defined functions are a powerful tool for modeling complex systems. They can be used in a variety of real-world applications, including physics, engineering, and economics. By understanding how to graph, evaluate, and use piecewise defined functions, you can gain a deeper understanding of these functions and their applications.

Final Answer

The final answer is:

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.
• To graph a piecewise defined function, you need to graph each sub-function on its respective interval.
• Piecewise defined functions can be used in a variety of real-world applications, including physics, engineering, and economics.
• To evaluate a piecewise defined function at a given point, you need to identify the sub-function that is defined on the interval containing the point.
• Piecewise defined functions can be used in calculus to model complex systems.