Identifying the Point for an Open Circle

Oct 13, 2025 by windailys.com 41 views

Understanding the Function f(x)

The given function f(x) is a piecewise function, which means it has different definitions for different intervals of x. In this case, the function is defined as:

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

This function has two parts: one for x less than 0 and another for x greater than or equal to 0.

Graphing the Function f(x)

To graph the function f(x), we need to consider both parts of the function. For x less than 0, the function is defined as -x, which means it is a decreasing linear function. For x greater than or equal to 0, the function is defined as 1, which means it is a constant function.

Identifying the Point for an Open Circle

The question asks at what point an open circle should be drawn. This typically indicates a point where the function is not defined or is not continuous. In this case, we need to examine the function at x = 0, as it is the boundary between the two parts of the function.

Analyzing the Function at x = 0

When x = 0, the function f(x) is defined as 1, which is a constant value. However, the function -x is not defined at x = 0, as it would result in division by zero. Therefore, the function is not continuous at x = 0.

Conclusion

Based on the analysis, the point where an open circle should be drawn is at x = 0, with a y-coordinate of 1. This is because the function is not continuous at x = 0, and an open circle is used to indicate a point where the function is not defined or is not continuous.

Answer

The correct answer is (0, 1).

Additional Considerations

It's worth noting that the function f(x) can be graphed on a coordinate plane using the two parts of the function. The graph would consist of a decreasing linear function for x less than 0 and a constant function for x greater than or equal to 0. The point (0, 1) would be marked with an open circle to indicate that the function is not continuous at x = 0.

Graphing the Function f(x)

Here is a step-by-step guide to graphing the function f(x):

Graph the decreasing linear function -x for x less than 0.
Graph the constant function 1 for x greater than or equal to 0.
Mark the point (0, 1) with an open circle to indicate that the function is not continuous at x = 0.

Conclusion

Q: What is the function f(x) and how is it defined?

A: The function f(x) is a piecewise function, which means it has different definitions for different intervals of x. In this case, the function is defined as:

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

This function has two parts: one for x less than 0 and another for x greater than or equal to 0.

Q: How do I graph the function f(x)?

A: To graph the function f(x), you need to consider both parts of the function. For x less than 0, the function is defined as -x, which means it is a decreasing linear function. For x greater than or equal to 0, the function is defined as 1, which means it is a constant function.

Q: What is the point where an open circle should be drawn?

A: The point where an open circle should be drawn is at x = 0, with a y-coordinate of 1. This is because the function is not continuous at x = 0, and an open circle is used to indicate a point where the function is not defined or is not continuous.

Q: Why is the function f(x) not continuous at x = 0?

A: The function f(x) is not continuous at x = 0 because the function -x is not defined at x = 0, as it would result in division by zero. Therefore, the function is not continuous at x = 0.

Q: How do I mark the point (0, 1) on the graph?

A: To mark the point (0, 1) on the graph, you should use an open circle. This indicates that the function is not continuous at x = 0.

Q: What is the significance of using an open circle to mark the point (0, 1)?

A: Using an open circle to mark the point (0, 1) indicates that the function is not continuous at x = 0. This is an important concept in graphing functions, as it helps to identify points where the function may not be defined or may not be continuous.

Q: Can I use a closed circle to mark the point (0, 1)?

A: No, you should not use a closed circle to mark the point (0, 1). A closed circle is used to indicate a point where the function is continuous, whereas an open circle is used to indicate a point where the function is not continuous.

Q: How do I determine whether a function is continuous or not?

A: To determine whether a function is continuous or not, you need to examine the function at the point in question. If the function is defined and has a finite value at the point, then the function is continuous at that point. If the function is not defined or has an infinite value at the point, then the function is not continuous at that point.

Q: What are some common types of discontinuities in functions?

A: Some common types of discontinuities in functions include:

Removable discontinuity: This occurs when a function is not defined at a point, but the limit of the function as x approaches that point exists.
Jump discontinuity: This occurs when a function has different values on either side of a point, but the limit of the function as x approaches that point does not exist.
Infinite discontinuity: This occurs when a function has an infinite value at a point, or when the limit of the function as x approaches that point is infinite.

Q: How do I graph a function with a removable discontinuity?

A: To graph a function with a removable discontinuity, you should mark the point where the discontinuity occurs with an open circle. This indicates that the function is not continuous at that point.

Q: How do I graph a function with a jump discontinuity?

A: To graph a function with a jump discontinuity, you should mark the point where the discontinuity occurs with an open circle. This indicates that the function is not continuous at that point.

Q: How do I graph a function with an infinite discontinuity?

A: To graph a function with an infinite discontinuity, you should mark the point where the discontinuity occurs with an open circle. This indicates that the function is not continuous at that point.

Conclusion

In conclusion, the function f(x) is a piecewise function that has different definitions for different intervals of x. The function is not continuous at x = 0, and an open circle should be drawn at the point (0, 1) to indicate this discontinuity.