The Function F ( X F(x F ( X ] Is To Be Graphed On A Coordinate Plane:${ F(x) = \begin{cases} -x, & X \ \textless \ 0 \ 1, & X \geq 0 \end{cases} }$At What Point Should An Open Circle Be Drawn?A. ( − 1 , 0 (-1,0 ( − 1 , 0 ]B. ( 0 , 0 (0,0 ( 0 , 0 ]C.

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Introduction

In this article, we will delve into the world of mathematical functions and explore the graphical representation of the function $f(x)$. The function $f(x)$ is defined as a piecewise function, with two different rules for $x < 0$ and $x \geq 0$. Our goal is to determine the point at which an open circle should be drawn on the coordinate plane.

The Piecewise Function $f(x)$

The function $f(x)$ is defined as:

$$f(x) = \begin{cases} -x, & x < 0 \\ 1, & x \geq 0 \end{cases}$$

This means that for any value of $x$ less than 0, the function $f(x)$ will take on the value of $-x$. On the other hand, for any value of $x$ greater than or equal to 0, the function $f(x)$ will take on the value of 1.

Graphing the Function $f(x)$

To graph the function $f(x)$, we need to consider two cases: $x < 0$ and $x \geq 0$.

Case 1: $x < 0$

For $x < 0$, the function $f(x)$ takes on the value of $-x$. This means that as $x$ decreases, the value of $f(x)$ will increase. We can graph this by plotting a line with a negative slope.

Case 2: $x \geq 0$

For $x \geq 0$, the function $f(x)$ takes on the value of 1. This means that for all values of $x$ greater than or equal to 0, the function $f(x)$ will be constant at 1. We can graph this by plotting a horizontal line at $y = 1$.

Determining the Point for an Open Circle

Now that we have graphed the function $f(x)$, we need to determine the point at which an open circle should be drawn. An open circle is used to indicate a point that is not included in the graph.

Looking at the graph, we can see that there is a discontinuity at $x = 0$. This means that the function $f(x)$ is not defined at $x = 0$. Therefore, an open circle should be drawn at the point $(0, 1)$.

Conclusion

In conclusion, the function $f(x)$ is a piecewise function that is defined differently for $x < 0$ and $x \geq 0$. We have graphed the function $f(x)$ and determined that an open circle should be drawn at the point $(0, 1)$.

Answer

The correct answer is:

• B. $(0,0)$

This is because the function $f(x)$ is not defined at $x = 0$, and an open circle should be drawn at the point $(0, 1)$.

Discussion

The function $f(x)$ is a classic example of a piecewise function. It is defined differently for $x < 0$ and $x \geq 0$, and it has a discontinuity at $x = 0$. This makes it a great example for illustrating the concept of piecewise functions and discontinuities.

Additional Examples

Here are a few additional examples of piecewise functions:

• $f(x) = \begin{cases} 2x, & x < 0 \\ 3, & x \geq 0 \end{cases}$
• $f(x) = \begin{cases} x^2, & x < 0 \\ 4, & x \geq 0 \end{cases}$

These functions can be graphed and analyzed in a similar way to the function $f(x)$.

Conclusion

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Introduction

In our previous article, we explored the graphical representation of the function $f(x)$. The function $f(x)$ is defined as a piecewise function, with two different rules for $x < 0$ and $x \geq 0$. We determined that an open circle should be drawn at the point $(0, 1)$. In this article, we will answer some frequently asked questions about the function $f(x)$.

Q&A

Q: What is the domain of the function $f(x)$?

A: The domain of the function $f(x)$ is all real numbers, except for $x = 0$. This is because the function $f(x)$ is not defined at $x = 0$.

Q: What is the range of the function $f(x)$?

A: The range of the function $f(x)$ is all real numbers, except for $y = 0$. This is because the function $f(x)$ takes on the value of $-x$ for $x < 0$, and the value of 1 for $x \geq 0$.

Q: Is the function $f(x)$ continuous?

A: No, the function $f(x)$ is not continuous. It has a discontinuity at $x = 0$.

Q: What is the value of the function $f(x)$ at $x = -1$?

A: The value of the function $f(x)$ at $x = -1$ is $1$. This is because $-1 < 0$, so the function $f(x)$ takes on the value of $-x$, which is $-(-1) = 1$.

Q: What is the value of the function $f(x)$ at $x = 1$?

A: The value of the function $f(x)$ at $x = 1$ is $1$. This is because $1 \geq 0$, so the function $f(x)$ takes on the value of 1.

Q: Can we simplify the function $f(x)$?

A: Yes, we can simplify the function $f(x)$ by writing it as:

$$f(x) = \begin{cases} -x, & x < 0 \\ 1, & x \geq 0 \end{cases} = \begin{cases} -x, & x < 0 \\ 1, & x \geq 0 \end{cases}$$

This is because the two cases are mutually exclusive, and the function $f(x)$ takes on the value of $-x$ for $x < 0$, and the value of 1 for $x \geq 0$.

Conclusion

In conclusion, the function $f(x)$ is a piecewise function that is defined differently for $x < 0$ and $x \geq 0$. We have answered some frequently asked questions about the function $f(x)$, and we have simplified the function $f(x)$ by writing it as:

$$f(x) = \begin{cases} -x, & x < 0 \\ 1, & x \geq 0 \end{cases}$$

This is a classic example of a piecewise function and discontinuity, and it is a great example for illustrating these concepts.

Additional Resources

• Graphing Piecewise Functions
• Discontinuities in Piecewise Functions
• Simplifying Piecewise Functions

Conclusion

In conclusion, the function $f(x)$ is a piecewise function that is defined differently for $x < 0$ and $x \geq 0$. We have answered some frequently asked questions about the function $f(x)$, and we have simplified the function $f(x)$ by writing it as:

$$f(x) = \begin{cases} -x, & x < 0 \\ 1, & x \geq 0 \end{cases}$$

This is a classic example of a piecewise function and discontinuity, and it is a great example for illustrating these concepts.