Understanding the Domain of a Square Root Function

Oct 13, 2025 by windailys.com 113 views

Introduction

When dealing with functions that involve square roots, it's essential to consider the domain of the function. The domain of a function is the set of all possible input values for which the function is defined. In this article, we will determine whether each given $x$ -value is in the domain of the function $f(x) = \sqrt{-2x}$ or not.

Understanding the Domain of a Square Root Function

The square root function is defined only for non-negative real numbers. This means that the expression inside the square root must be greater than or equal to zero. In the case of the function $f(x) = \sqrt{-2x}$ , the expression inside the square root is $-2x$ . For this expression to be non-negative, we must have $-2x \geq 0$ .

Solving the Inequality

To solve the inequality $-2x \geq 0$ , we can divide both sides by $-2$ . However, when we divide by a negative number, we must reverse the direction of the inequality sign. Therefore, we have:

x \leq 0

This means that the domain of the function $f(x) = \sqrt{-2x}$ consists of all real numbers $x$ such that $x \leq 0$ .

Evaluating the Given $x$ -Values

Now that we have determined the domain of the function, we can evaluate each given $x$ -value to see whether it is in the domain or not.

A. $x = -2$

Since $x = -2$ is less than or equal to zero, it is in the domain of the function.

B. $x = 2$

Since $x = 2$ is greater than zero, it is not in the domain of the function.

C. $x = 4$

Since $x = 4$ is greater than zero, it is not in the domain of the function.

Conclusion

In conclusion, the domain of the function $f(x) = \sqrt{-2x}$ consists of all real numbers $x$ such that $x \leq 0$ . We have evaluated each given $x$ -value and determined whether it is in the domain or not. The results are as follows:

$x = -2$ is in the domain.
$x = 2$ is not in the domain.
$x = 4$ is not in the domain.

Domain of a Square Root Function

The domain of a square root function is the set of all possible input values for which the function is defined. In the case of the function $f(x) = \sqrt{-2x}$ , the domain consists of all real numbers $x$ such that $x \leq 0$ .

Key Takeaways

The domain of a square root function is the set of all possible input values for which the function is defined.
The expression inside the square root must be non-negative for the function to be defined.
The domain of the function $f(x) = \sqrt{-2x}$ consists of all real numbers $x$ such that $x \leq 0$ .

Examples of Square Root Functions

$f(x) = \sqrt{x}$
$f(x) = \sqrt{-x}$
$f(x) = \sqrt{2x}$

Solving Inequalities

Solving the inequality $-2x \geq 0$ gives us $x \leq 0$ .
Solving the inequality $x \geq 0$ gives us $x \geq 0$ .

Domain of a Function

The domain of a function is the set of all possible input values for which the function is defined.
The domain of a function can be determined by analyzing the expression inside the square root.

Conclusion

$x = -2$ is in the domain.
$x = 2$ is not in the domain.
$x = 4$ is not in the domain.
Frequently Asked Questions (FAQs) About the Domain of a Square Root Function ================================================================================

Q: What is the domain of a square root function?

A: The domain of a square root function is the set of all possible input values for which the function is defined. In the case of the function $f(x) = \sqrt{-2x}$ , the domain consists of all real numbers $x$ such that $x \leq 0$ .

Q: Why is the domain of a square root function important?

A: The domain of a square root function is important because it determines the set of all possible input values for which the function is defined. If the input value is not in the domain, the function is not defined.

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

A: To determine the domain of a square root function, you need to analyze the expression inside the square root. The expression inside the square root must be non-negative for the function to be defined.

Q: What happens if the expression inside the square root is negative?

A: If the expression inside the square root is negative, the function is not defined. This is because the square root of a negative number is not a real number.

Q: Can I have a negative value inside the square root?

A: Yes, you can have a negative value inside the square root, but the expression inside the square root must be non-negative for the function to be defined.

Q: How do I solve the inequality $-2x \geq 0$ ?

A: To solve the inequality $-2x \geq 0$ , you can divide both sides by $-2$ . However, when you divide by a negative number, you must reverse the direction of the inequality sign. Therefore, you have $x \leq 0$ .

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

A: The domain of the function $f(x) = \sqrt{x}$ consists of all real numbers $x$ such that $x \geq 0$ .

Q: What is the domain of the function $f(x) = \sqrt{-x}$ ?

A: The domain of the function $f(x) = \sqrt{-x}$ consists of all real numbers $x$ such that $x \leq 0$ .

Q: Can I have a fraction inside the square root?

A: Yes, you can have a fraction inside the square root, but the expression inside the square root must be non-negative for the function to be defined.

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 analyze the expression inside the square root. The expression inside the square root must be non-negative for the function to be defined.

Q: What is the domain of the function $f(x) = \sqrt{2x}$ ?

A: The domain of the function $f(x) = \sqrt{2x}$ consists of all real numbers $x$ such that $x \geq 0$ .

Conclusion

In conclusion, the domain of a square root function is the set of all possible input values for which the function is defined. The expression inside the square root must be non-negative for the function to be defined. We have answered some frequently asked questions about the domain of a square root function and provided examples of functions with a square root.