Domain and Range of a Square Root Function

Oct 13, 2025 by miodeya.com 102 views

Introduction

When dealing with functions, it's essential to understand the domain and range of the function. The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values. In this article, we will explore the domain and range of the function $f(x)=\sqrt{-x}$ .

The Square Root Function

The square root function is defined as $f(x)=\sqrt{x}$ , where $x$ is a real number. However, when we have a negative value inside the square root, such as $-x$ , we need to consider the domain and range of the function carefully.

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 function $f(x)=\sqrt{-x}$ , we need to consider the values of $x$ that make the expression inside the square root negative.

Since $-x$ is always negative, the domain of the function is all real numbers less than or equal to 0. This is because any real number $x$ that is less than or equal to 0 will result in a negative value for $-x$ , which can be taken as the input for the square root function.

Range of the Function

The range of a function is the set of all possible output values. In the case of the function $f(x)=\sqrt{-x}$ , we need to consider the values of $f(x)$ that can be obtained.

Since the square root of a negative number is an imaginary number, the range of the function is not all real numbers. Instead, the range of the function is the set of all non-negative imaginary numbers.

Conclusion

In conclusion, the domain of the function $f(x)=\sqrt{-x}$ is all real numbers less than or equal to 0, while the range of the function is the set of all non-negative imaginary numbers.

Answer

Based on the analysis above, the correct answer is:

C. The domain of the graph is all real numbers less than or equal to 0.

Explanation

The domain of the function $f(x)=\sqrt{-x}$ is all real numbers less than or equal to 0 because any real number $x$ that is less than or equal to 0 will result in a negative value for $-x$ , which can be taken as the input for the square root function.

Final Thoughts

Understanding the domain and range of a function is crucial in mathematics, especially when dealing with functions that involve square roots or other operations that can result in imaginary numbers. By carefully analyzing the domain and range of a function, we can gain a deeper understanding of the function's behavior and make informed decisions when working with mathematical expressions.

Domain and Range of a Square Root Function

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 $f(x)=\sqrt{x}$ , the domain is all real numbers greater than or equal to 0.

Range of a Square Root Function

The range of a square root function is the set of all possible output values. In the case of the function $f(x)=\sqrt{x}$ , the range is all non-negative real numbers.

Domain and Range of a Negative Square Root Function

When we have a negative value inside the square root, such as $-x$ , we need to consider the domain and range of the function carefully. The domain of the function $f(x)=\sqrt{-x}$ is all real numbers less than or equal to 0, while the range of the function is the set of all non-negative imaginary numbers.

Example Problems

Problem 1

Find the domain and range of the function $f(x)=\sqrt{2x}$ .

Domain: The domain of the function is all real numbers greater than or equal to 0.
Range: The range of the function is all non-negative real numbers.

Problem 2

Find the domain and range of the function $f(x)=\sqrt{-2x}$ .

Domain: The domain of the function is all real numbers less than or equal to 0.
Range: The range of the function is the set of all non-negative imaginary numbers.

Conclusion

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

A: The domain of the function $f(x)=\sqrt{x}$ is all real numbers greater than or equal to 0.

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

A: The range of the function $f(x)=\sqrt{x}$ is all non-negative real numbers.

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

A: The domain of the function $f(x)=\sqrt{-x}$ is all real numbers less than or equal to 0.

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

A: The range of the function $f(x)=\sqrt{-x}$ is the set of all non-negative imaginary numbers.

Q: Can the square root of a negative number be a real number?

A: No, the square root of a negative number cannot be a real number. The square root of a negative number is an imaginary number.

Q: Can the square root of a negative number be a non-negative real number?

A: No, the square root of a negative number cannot be a non-negative real number. The square root of a negative number is an imaginary number.

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

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

Q: Why is it important to understand the domain and range of a function?

A: Understanding the domain and range of a function is crucial in mathematics, especially when dealing with functions that involve square roots or other operations that can result in imaginary numbers. By carefully analyzing the domain and range of a function, we can gain a deeper understanding of the function's behavior and make informed decisions when working with mathematical expressions.

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

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

Q: Can the domain of a function be empty?

A: Yes, the domain of a function can be empty. For example, the function $f(x)=\sqrt{x}$ has an empty domain when $x$ is less than 0.

Q: Can the range of a function be empty?

A: Yes, the range of a function can be empty. For example, the function $f(x)=x^2+1$ has an empty range when $x$ is a real number.

Conclusion

In conclusion, understanding the domain and range of a function is crucial in mathematics, especially when dealing with functions that involve square roots or other operations that can result in imaginary numbers. By carefully analyzing the domain and range of a function, we can gain a deeper understanding of the function's behavior and make informed decisions when working with mathematical expressions.

Additional Resources

Final Thoughts

Understanding the domain and range of a function is a fundamental concept in mathematics, and it's essential to have a clear understanding of these concepts when working with functions. By carefully analyzing the domain and range of a function, we can gain a deeper understanding of the function's behavior and make informed decisions when working with mathematical expressions.