Select The Correct Answer.Which Statement Is True About The Given Function? F ( X ) = 6 X 2 + 1 F(x)=\frac{6}{x^2+1} F ( X ) = X 2 + 1 6 A. Function F F F Is Continuous And Has A Domain Of ( 0 , ∞ (0, \infty ( 0 , ∞ ].B. Function F F F Is Discontinuous And Has A Domain Of

Mar 7, 2025 by ADMIN 275 views

**Evaluating the Continuity and Domain of a Function**

Understanding the Function

The given function is $f(x)=\frac{6}{x^2+1}$ . This is a rational function, where the numerator is a constant and the denominator is a quadratic expression. To determine the continuity and domain of this function, we need to analyze its behavior as $x$ varies.

Continuity of the Function

A function is said to be continuous at a point $x=a$ if the following conditions are met:

The function is defined at $x=a$ .
The limit of the function as $x$ approaches $a$ exists.
The limit of the function as $x$ approaches $a$ is equal to the value of the function at $x=a$ .

In the case of the given function, we can see that the denominator $x^2+1$ is always positive, since it is a sum of squares. Therefore, the function is defined for all real values of $x$ . Moreover, the function is a rational function, and the numerator and denominator are both polynomials, which means that the function is continuous everywhere.

Domain of the Function

The domain of a function is the set of all possible input values for which the function is defined. In the case of the given function, we have already established that the function is defined for all real values of $x$ . Therefore, the domain of the function is the set of all real numbers, which can be represented as $(-\infty, \infty)$ .

Comparing the Options

Now that we have analyzed the continuity and domain of the given function, we can compare the options provided:

A. Function $f$ is continuous and has a domain of $(0, \infty)$ .

B. Function $f$ is discontinuous and has a domain of $(-\infty, \infty)$ .

Based on our analysis, we can see that option A is incorrect, since the function is continuous everywhere, not just in the interval $(0, \infty)$ . Option B is also incorrect, since the function is continuous everywhere, not discontinuous.

Conclusion

In conclusion, the correct statement about the given function is that it is continuous and has a domain of $(-\infty, \infty)$ . This means that the function is defined for all real values of $x$ , and it is continuous everywhere.

Key Takeaways

A function is continuous at a point $x=a$ if the function is defined at $x=a$ , the limit of the function as $x$ approaches $a$ exists, and the limit of the function as $x$ approaches $a$ is equal to the value of the function at $x=a$ .
The domain of a function is the set of all possible input values for which the function is defined.
A rational function is continuous everywhere, since the numerator and denominator are both polynomials.

Final Answer

The correct answer is:

C. Function $f$ is continuous and has a domain of $(-\infty, \infty)$ .

Q: What is continuity in a function?

A: Continuity in a function refers to the ability of the function to be defined at a point, and the limit of the function as it approaches that point exists and is equal to the value of the function at that point.

Q: What are the conditions for a function to be continuous?

A: A function is continuous at a point $x=a$ if the following conditions are met:

The function is defined at $x=a$ .
The limit of the function as $x$ approaches $a$ exists.
The limit of the function as $x$ approaches $a$ is equal to the value of the function at $x=a$ .

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined.

Q: How do you determine the domain of a function?

A: To determine the domain of a function, you need to identify any values of $x$ that would make the denominator of the function equal to zero, and exclude those values from the domain.

Q: What is the difference between a rational function and an irrational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials, while an irrational function is a function that cannot be expressed as the ratio of two polynomials.

Q: Is a rational function always continuous?

A: Yes, a rational function is always continuous, since the numerator and denominator are both polynomials, and the function is defined for all real values of $x$ .

Q: What is the domain of a rational function?

A: The domain of a rational function is the set of all real numbers, unless there are any values of $x$ that would make the denominator equal to zero, in which case those values are excluded from the domain.

Q: Can a rational function have a discontinuity?

A: No, a rational function cannot have a discontinuity, since it is defined for all real values of $x$ and the limit of the function as $x$ approaches any point exists and is equal to the value of the function at that point.

Q: How do you determine if a function is continuous or discontinuous?

A: To determine if a function is continuous or discontinuous, you need to check if the function is defined at a point, and if the limit of the function as it approaches that point exists and is equal to the value of the function at that point.

Q: What is the significance of continuity in a function?

A: Continuity in a function is significant because it allows us to use the properties of limits to analyze the behavior of the function, and to make conclusions about the function's behavior at different points.

Q: Can a function be continuous at a single point?

A: Yes, a function can be continuous at a single point, but it is more common for a function to be continuous over an interval or a set of points.

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

A: A continuous function is a function that is continuous over its entire domain, while a piecewise continuous function is a function that is continuous over different intervals or sets of points.

Q: Can a piecewise continuous function be continuous at a single point?

A: Yes, a piecewise continuous function can be continuous at a single point, but it is more common for a piecewise continuous function to be continuous over different intervals or sets of points.

Q: How do you determine if a piecewise continuous function is continuous at a single point?

A: To determine if a piecewise continuous function is continuous at a single point, you need to check if the function is defined at that point, and if the limit of the function as it approaches that point exists and is equal to the value of the function at that point.