What Is The Domain Of The Function $y=\sqrt{x}$?A. $-\infty\ \textless \ X\ \textless \ \infty$B. $0\ \textless \ X\ \textless \ \infty$C. $0 \leq X\ \textless \ \infty$D. $1 \leq X\ \textless \ \infty$

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Understanding the Concept of Domain

The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of the variable that the function can accept without resulting in an undefined or imaginary output. In this article, we will explore the concept of domain and determine the domain of the function $y=\sqrt{x}$.

What is the Function $y=\sqrt{x}$?

The function $y=\sqrt{x}$ is a square root function, which is defined as the inverse of the square function. It is a mathematical operation that takes a number as input and returns a value that, when multiplied by itself, gives the original number. The square root function is denoted by the symbol $\sqrt{x}$ and is defined for all non-negative real numbers.

Understanding the Domain of the Function $y=\sqrt{x}$

To determine the domain of the function $y=\sqrt{x}$, we need to consider the values of $x$ for which the function is defined. Since the square root function is defined for all non-negative real numbers, the domain of the function $y=\sqrt{x}$ must include all non-negative real numbers.

Analyzing the Options

Let's analyze the options given to determine the correct domain of the function $y=\sqrt{x}$.

Option A: $-\infty\ \textless \ x\ \textless \ \infty$

This option suggests that the domain of the function $y=\sqrt{x}$ includes all real numbers, both positive and negative. However, since the square root function is defined only for non-negative real numbers, this option is incorrect.

Option B: $0\ \textless \ x\ \textless \ \infty$

This option suggests that the domain of the function $y=\sqrt{x}$ includes all positive real numbers. However, since the square root function is defined for all non-negative real numbers, this option is also incorrect.

Option C: $0 \leq x\ \textless \ \infty$

This option suggests that the domain of the function $y=\sqrt{x}$ includes all non-negative real numbers. Since the square root function is defined for all non-negative real numbers, this option is correct.

Option D: $1 \leq x\ \textless \ \infty$

This option suggests that the domain of the function $y=\sqrt{x}$ includes all real numbers greater than or equal to 1. However, since the square root function is defined for all non-negative real numbers, this option is incorrect.

Conclusion

In conclusion, the domain of the function $y=\sqrt{x}$ is the set of all non-negative real numbers, which can be represented as $0 \leq x\ \textless \ \infty$. This means that the function $y=\sqrt{x}$ is defined for all values of $x$ that are greater than or equal to 0 and less than infinity.

Final Answer

The final answer is C.

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Q&A: Domain of the Function $y=\sqrt{x}$

In this article, we will answer some frequently asked questions about the domain of the function $y=\sqrt{x}$.

Q: What is the domain of the function $y=\sqrt{x}$?

A: The domain of the function $y=\sqrt{x}$ is the set of all non-negative real numbers, which can be represented as $0 \leq x\ \textless \ \infty$.

Q: Why is the domain of the function $y=\sqrt{x}$ limited to non-negative real numbers?

A: The domain of the function $y=\sqrt{x}$ is limited to non-negative real numbers because the square root function is defined only for non-negative real numbers. When you take the square root of a negative number, you get an imaginary number, which is not a real number.

Q: Can the domain of the function $y=\sqrt{x}$ be extended to include negative real numbers?

A: No, the domain of the function $y=\sqrt{x}$ cannot be extended to include negative real numbers. The square root function is defined only for non-negative real numbers, and it is not possible to extend its domain to include negative real numbers.

Q: What happens if you try to take the square root of a negative number?

A: If you try to take the square root of a negative number, you will get an imaginary number. For example, $\sqrt{-1}$ is equal to $i$, which is an imaginary number.

Q: Can the domain of the function $y=\sqrt{x}$ be extended to include complex numbers?

A: Yes, the domain of the function $y=\sqrt{x}$ can be extended to include complex numbers. However, this is a more advanced topic in mathematics, and it requires a good understanding of complex numbers and their properties.

Q: What is the significance of the domain of the function $y=\sqrt{x}$?

A: The domain of the function $y=\sqrt{x}$ is significant because it determines the set of input values for which the function is defined. In other words, it determines the set of values of $x$ for which the function $y=\sqrt{x}$ is a valid mathematical operation.

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

A: To determine the domain of a function, you need to consider the values of the input variable that make the function undefined or imaginary. In the case of the function $y=\sqrt{x}$, the domain is limited to non-negative real numbers because the square root function is defined only for non-negative real numbers.

Q: Can the domain of a function be changed?

A: No, the domain of a function cannot be changed. The domain of a function is a fixed set of values that determine the set of input values for which the function is defined.

Q: What is the relationship between the domain and the range of a function?

A: The domain and the range of a function are related in the sense that the domain determines the set of input values for which the function is defined, and the range determines the set of output values that the function produces.

Q: Can the domain and the range of a function be the same?

A: Yes, the domain and the range of a function can be the same. For example, the function $y=x$ has the same domain and range, which is the set of all real numbers.

Q: What is the significance of the domain and the range of a function?

A: The domain and the range of a function are significant because they determine the set of input values and the set of output values for which the function is defined. In other words, they determine the set of values of $x$ and $y$ for which the function is a valid mathematical operation.

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

A: To determine the range of a function, you need to consider the set of output values that the function produces for each input value in the domain. In other words, you need to consider the set of values of $y$ that correspond to each value of $x$ in the domain.

Q: Can the range of a function be changed?

A: No, the range of a function cannot be changed. The range of a function is a fixed set of values that determine the set of output values that the function produces.

Q: What is the relationship between the domain and the range of a function?

A: The domain and the range of a function are related in the sense that the domain determines the set of input values for which the function is defined, and the range determines the set of output values that the function produces.

Q: Can the domain and the range of a function be the same?

A: Yes, the domain and the range of a function can be the same. For example, the function $y=x$ has the same domain and range, which is the set of all real numbers.

Q: What is the significance of the domain and the range of a function?

A: The domain and the range of a function are significant because they determine the set of input values and the set of output values for which the function is defined. In other words, they determine the set of values of $x$ and $y$ for which the function is a valid mathematical operation.

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

A: To determine the domain and the range of a function, you need to consider the set of input values and the set of output values that the function produces. In other words, you need to consider the set of values of $x$ and $y$ for which the function is a valid mathematical operation.

Q: Can the domain and the range of a function be changed?

A: No, the domain and the range of a function cannot be changed. The domain and the range of a function are fixed sets of values that determine the set of input values and the set of output values for which the function is defined.

Q: What is the relationship between the domain and the range of a function?

A: The domain and the range of a function are related in the sense that the domain determines the set of input values for which the function is defined, and the range determines the set of output values that the function produces.

Q: Can the domain and the range of a function be the same?

A: Yes, the domain and the range of a function can be the same. For example, the function $y=x$ has the same domain and range, which is the set of all real numbers.

Q: What is the significance of the domain and the range of a function?

A: The domain and the range of a function are significant because they determine the set of input values and the set of output values for which the function is defined. In other words, they determine the set of values of $x$ and $y$ for which the function is a valid mathematical operation.

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

A: To determine the domain and the range of a function, you need to consider the set of input values and the set of output values that the function produces. In other words, you need to consider the set of values of $x$ and $y$ for which the function is a valid mathematical operation.

Q: Can the domain and the range of a function be changed?

A: No, the domain and the range of a function cannot be changed. The domain and the range of a function are fixed sets of values that determine the set of input values and the set of output values for which the function is defined.

Q: What is the relationship between the domain and the range of a function?

A: The domain and the range of a function are related in the sense that the domain determines the set of input values for which the function is defined, and the range determines the set of output values that the function produces.

Q: Can the domain and the range of a function be the same?

A: Yes, the domain and the range of a function can be the same. For example, the function $y=x$ has the same domain and range, which is the set of all real numbers.

Q: What is the significance of the domain and the range of a function?

A: The domain and the range of a function are significant because they determine the set of input values and the set of output values for which the function is defined. In other words, they determine the set of values of $x$ and $y$ for which the function is a valid mathematical operation.

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

A: To determine the domain and the range of a function, you need to consider the set of input values and the set of output values that the function produces. In other words, you need to consider the set of values of $x$ and $y$ for which the function is a valid mathematical operation.

Q: Can the domain and the range of a function be changed?

A: No, the domain and the range of a function cannot be changed. The domain and the range of a function are fixed sets of values that determine the set of input values and the set of output values for which the function is defined.

Q: What is the relationship between the domain and the range of a function?

A: The domain and the range of a function are related in the sense that the domain determines the set of input values for which the function is defined, and the range determines the set of output values that the function produces.

Q: Can the domain and the range of a function be the same?

A: Yes, the domain and the range of a function can be the same. For