What Is The Domain Of The Function Y = 2 X − 6 Y = 2 \sqrt{x-6} Y = 2 X − 6 ​ ?A. − ∞ \textless X \textless ∞ -\infty \ \textless \ X \ \textless \ \infty − ∞ \textless X \textless ∞ B. 0 ≤ X \textless ∞ 0 \leq X \ \textless \ \infty 0 ≤ X \textless ∞ C. 3 ≤ X \textless ∞ 3 \leq X \ \textless \ \infty 3 ≤ X \textless ∞ D. $6 \leq X \ \textless \

Introduction

When dealing with functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output. In this article, we'll focus on finding the domain of the function $y = 2 \sqrt{x-6}$.

What is the Domain of a Function?

The domain of a function is a critical concept in mathematics, particularly in algebra and calculus. It's essential to understand that the domain of a function is not just a set of numbers, but rather a set of values that the function can accept without resulting in an undefined or imaginary output.

Why is the Domain Important?

The domain of a function is important because it determines the range of values that the function can produce. If the domain of a function is not properly defined, it can lead to incorrect results and even affect the accuracy of mathematical models.

Finding the Domain of the Function $y = 2 \sqrt{x-6}$

To find the domain of the function $y = 2 \sqrt{x-6}$, we need to consider the restrictions imposed by the square root function. The square root function is only defined for non-negative values, which means that the expression inside the square root must be greater than or equal to zero.

Step 1: Identify the Restrictions Imposed by the Square Root Function

The square root function is only defined for non-negative values, which means that the expression inside the square root must be greater than or equal to zero. In this case, the expression inside the square root is $x-6$. Therefore, we need to find the values of x for which $x-6 \geq 0$.

Step 2: Solve the Inequality $x-6 \geq 0$

To solve the inequality $x-6 \geq 0$, we need to isolate the variable x. We can do this by adding 6 to both sides of the inequality, which gives us $x \geq 6$.

Step 3: Consider the Domain of the Function

Now that we have found the values of x for which $x-6 \geq 0$, we need to consider the domain of the function. Since the function is defined for all values of x that satisfy the inequality $x \geq 6$, the domain of the function is all real numbers greater than or equal to 6.

Conclusion

In conclusion, the domain of the function $y = 2 \sqrt{x-6}$ is all real numbers greater than or equal to 6. This means that the function is defined for all values of x that satisfy the inequality $x \geq 6$. We can represent this domain using interval notation as $[6, \infty)$.

Answer

The correct answer is:

• C. $3 \leq x \ \textless \ \infty$ is incorrect because the domain of the function is all real numbers greater than or equal to 6, not 3.
• D. $6 \leq x \ \textless \ \infty$ is correct because the domain of the function is all real numbers greater than or equal to 6.

Final Answer

Introduction

In our previous article, we discussed the concept of the domain of a function and how to find the domain of the function $y = 2 \sqrt{x-6}$. In this article, we'll answer some frequently asked questions related to the domain of a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output.

Q: Why is the domain of a function important?

A: The domain of a function is important because it determines the range of values that the function can produce. If the domain of a function is not properly defined, it can lead to incorrect results and even affect the accuracy of mathematical models.

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

A: To find the domain of a function, you need to consider the restrictions imposed by the function. For example, if the function involves a square root, you need to ensure that the expression inside the square root is non-negative. You can also use interval notation to represent the domain of a function.

Q: What is interval notation?

A: Interval notation is a way of representing a set of numbers using a specific notation. For example, the set of all real numbers greater than or equal to 6 can be represented as $[6, \infty)$. Interval notation is commonly used to represent the domain and range of functions.

Q: How do I determine the domain of a function with a square root?

A: To determine the domain of a function with a square root, you need to ensure that the expression inside the square root is non-negative. You can do this by setting the expression inside the square root greater than or equal to zero and solving for the variable.

Q: What is the domain of the function $y = \sqrt{x^2 - 4}$?

A: To find the domain of the function $y = \sqrt{x^2 - 4}$, we need to ensure that the expression inside the square root is non-negative. We can do this by setting $x^2 - 4 \geq 0$ and solving for the variable. This gives us $x^2 \geq 4$, which means that $x \leq -2$ or $x \geq 2$. Therefore, the domain of the function is $(-\infty, -2] \cup [2, \infty)$.

Q: What is the domain of the function $y = \frac{1}{x-2}$?

A: To find the domain of the function $y = \frac{1}{x-2}$, we need to ensure that the denominator is not equal to zero. We can do this by setting $x-2 \neq 0$ and solving for the variable. This gives us $x \neq 2$. Therefore, the domain of the function is all real numbers except 2.

Conclusion

In conclusion, the domain of a function is a critical concept in mathematics that determines the range of values that the function can produce. By understanding the domain of a function, you can avoid incorrect results and ensure the accuracy of mathematical models. We hope that this article has helped you to better understand the domain of a function and how to find the domain of a function.

Frequently Asked Questions

• Q: What is the domain of a function? A: The domain of a function is the set of all possible input values (x-values) for which the function is defined.
• Q: Why is the domain of a function important? A: The domain of a function is important because it determines the range of values that the function can produce.
• Q: How do I find the domain of a function? A: To find the domain of a function, you need to consider the restrictions imposed by the function.
• Q: What is interval notation? A: Interval notation is a way of representing a set of numbers using a specific notation.

Final Answer

The final answer is that the domain of a function is a critical concept in mathematics that determines the range of values that the function can produce. By understanding the domain of a function, you can avoid incorrect results and ensure the accuracy of mathematical models.